ARS MATHEMATICA CONTEMPORANEA Volume 13, Number 1, Fall/Winter 2017, Pages 1-234 Covered by: Mathematical Reviews Zentralblatt MATH COBISS SCOPUS Science Citation Index-Expanded (SCIE) Web of Science ISI Alerting Service Current Contents/Physical, Chemical & Earth Sciences (CC/PC & ES) The University of Primorska The Society of Mathematicians, Physicists and Astronomers of Slovenia The Institute of Mathematics, Physics and Mechanics The publication is partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications. ARS MATHEMATICA CONTEMPORANEA Petra Šparl Award The Petra Šparl Award is for the best paper published in AMC or ADAM by a young woman mathematician, and is awarded by the Slovenian Society of Discrete and Applied Mathematics and University of Primorska each even year, beginning from 2018. 1. Nominations for the Award are sought through announcements in AMC and ADAM. 2. 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Dragan Marušic and Tomaž Pisanski Editors In Chief iii ARS MATHEMATICA CONTEMPORANEA Contents On the largest subsets avoiding the diameter of (0, ±1)-vectors Saori Adachi, Hiroshi Nozaki......................... 1 The distinguishing index of the Cartesian product of countable graphs Izak Broere, Monika Pilsniak......................... 15 Classification of convex polyhedra by their rotational orbit Euler characteristic Jurij Kovic...................................23 Lifting symmetric pictures to polyhedral scenes Viktčria E. Kaszanitzky, Bernd Schulze................... 31 On the Erdos-Sos Conjecture for graphs on n = k + 4 vertices Long-Tu Yuan, Xiao-Dong Zhang......................49 The JLS model with ARMA/GARCH errors Špela Jezernik Širca, Matjaž Omladic....................63 The existence of square integer Heffter arrays Jeffrey H. Dinitz, Ian M. Wanless....................... 81 3-pyramidal Steiner triple systems Marco Buratti, Gloria Rinaldi, Tommaso Traetta...............95 On some generalization of the Mobius configuration Krzysztof Petelczyc..............................107 Domination game on paths and cycles Gašper Košmrlj................................125 Affine primitive symmetric graphs of diameter two Carmen Amarra, Michael Giudici, Cheryl E. Praeger............137 On which groups can arise as the canonical group of a spherical latin bitrade Kyle Bonetta-Martin, Thomas A. McCourt..................167 Pursuit-evasion in a two-dimensional domain Andrew Beveridge, Yiqing Cai........................187 Spectrum, distance spectrum, and Wiener index of wreath products of complete graphs Alfredo Donno ................................ 207 Counting faces of graphical zonotopes Vladimir Grujic................................227 Volume 13, Number 1, Fall/Winter 2017, Pages 1-234 v /^creative ^commor ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 1-13 On the largest subsets avoiding the diameter of (0, ±1)-vectors Saori Adachi, Hiroshi Nozaki * Department of Mathematics Education, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi 448-8542, Japan Received 4 September 2015, accepted 28 October 2015, published online 25 July 2016 Abstract Let Lmkl c Rm+fc+' be the set of vectors which have m of entries -1, k of entries 0, and l of entries 1. In this paper, we investigate the largest subset of Lmkl whose diameter is smaller than that of Lmkl. The largest subsets for m =1, l = 2, and any k will be classified. From this result, we can classify the largest 4-distance sets containing the Euclidean representation of the Johnson scheme J(9,4). This was an open problem in Bannai, Sato, and Shigezumi (2012). Keywords: The ErdSs-Ko-Rado theorem, s-distance set, diameter graph, independent set, extremal set theory. Math. Subj. Class.: 05D05, 05C69 1 Introduction The famous theorem in Erdos element subsets of In = {1, then For n > 2k, the set {A c In | |A| = k, 1 G A} is the unique family achieving equality, up to permutations on In. For n = 2k, the largest set is any family which contains only one of A or In \ A for any k-element A c In. This result plays a central role in extremal set theory, and similar or analogous theorems are proved for various objects [2, 5, 9]. »Work supported by JSPS KAKENHI Grant Numbers 25800011, 26400003. E-mail addresses: s214m044@auecc.aichi-edu.ac.jp (Saori Adachi), hnozaki@auecc.aichi-edu.ac.jp (Hiroshi Nozaki) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ -Ko-Rado [8] stated that for n > 2k and a family A of k.., n}, if any two distinct A, B G A satisfy A n B = 0, |A| — (H 1 2 Ars Math. Contemp. 13 (2017) 15-21 We can naturally interpret A c In as x = (x^ ..., xn) G Rn by the manner xj = 1 if i G A, xj =0 if i G A. By this identification, the Erdos-Ko-Rado Theorem can be rewritten that for n > 2k and a subset X of Lk = {x G Rn | xj G {0,1}, ^ xj = k} if any distinct x, y G X satisfy d(x, y) < D(Lk) = %/2k, then where d(,) is the Euclidean distance, and D(Lk) is the diameter of Lk. We would like to consider the following problem to generalize the Erdos-Ko-Rado Theorem. Problem 1.1. Let Lmkl C Rm+fc+' be the set of vectors which have m of entries -1, k of entries 0, and l of entries 1. Classify the largest X c Lmkl with D(X) < D(LmkI). It is almost obvious for the cases m = l (Proposition 2.1) and m + k < l (Proposition 2.2). In this paper, we solve the first non-trivial case m =1, l = 2 and any k (Theorem 2.5). Using the largest sets for the case (m, k, l) = (1,6,2), we can classify the largest 4-distance sets containing the Euclidean representation of the Johnson scheme J(9,4). This was an open problem in [1]. We will give a brief survey on related results. Let Lnm be the set of (0, ±1)-vectors in Rn which have m non-zero coordinates. For a fixed set D of integers, let V(n, m, D) be the family of subsets V = {vi,..., vk } of Lnm such that (vj, vj) G D for any i = j. There are several results relating to the largest sets in V(n, m, D) for some (n, m, D) [4, 6, 7]. Since X c Lnm is on a sphere, if |D| = s holds, then |X| < (n+ss-i) + (d+--2) [3]. The case D = {d} is investigated in [4]. For non-negative integers d < m, t > 2, and n > n0(m) (see [4] about n0(m)), if X G V(n, m, {d, d + 1,..., d + t - 1}), then |X| < (n-d)/(m-d) [6]. This equality can be attained whenever a Steiner system S(n - d, m - d, t) (equivalently t-(n - d, m - d, 1) design) exists . We also have if X G V(n, m, {-(t - 1),-(t - 2),..., t - 1}),then |X| < 2t-i(m - t + !)(?)/([7]. When m = t +1, this equality can be attained whenever a Steiner system S(n, m, m - 1) 2 Largest subsets avoiding the diameter of Lmkl Let Lmkl denote the finite set in Rn = Rm+fc+', which consists of all vectors whose number of entries -1, 0,1 is equal to m, k, l, respectively. For two subsets X, Y of Lmkl, X is isomorphic to Y if there exists a permutation a G Sn such that X = {(yCT(i),..., yCT(n}) | (yi,..., yn) G Y}. The diameter D(X) of X c Rn is defined to be where d(,) is the Euclidean distance. Let Mmkl denote the largest possible number of cardinalities of X c Lmkl such that D(X) < D(LmkI). The diameter graph of X c Rn is defined to be the graph (X, E), where E = {(x, y) | d(x, y) = D(X)}. The problem of determining Mmkl is equivalent to determining the independence number of the diameter graph of Lmfci. Note that Mmfci = Mifcm because we have Lmfci = -Lifcm = {-x | x G ¿ifcm}. Thus we may assume m < l. In this section, we determine Mmkl, and classify the largest sets for several cases of m, k, l. First we determine Mmkl for the cases m = l and m + k < l. exists. D(X) = max{d(x, y) | x, y G X}, S. Adachi and H. Nozaki: On the largest subsets avoiding the diameter of (0, ±1)-vectors 3 Proposition 2.1. Assume m = I. Then we have 1f n\ik + m\ 1 Mmkl = - = - |Lmkl|, 2 \mj \ m J 2 and the largest sets contain only one of x or —x for any x G Lmkl. Proof. For any x G Lmkl, we have {y | d(x,y) = D(Lmkl)} = {—x}. Therefore the diameter graph of Lmkl is the set of independent edges. The proposition can be easily proved from this fact. □ For X c Lmkl, we use the notation Ni(X,j) = {(xi,...,xn) G X | xi = j}, and nm(X,j) = N(X,j)|. Proposition 2.2. Assume m + k < I. Then we have n — 1 \ f m + k^ Mmkl , , , 1 . . vm + k — 1/ y m For m + k > l, the largest set is Nl(Lmkl, —1) U Nl(Lmkl, 0), up to isomorphism. For m + k = l, then the largest sets contain only one of {(xi,..., xn) G Lmkl | xj = 1, V« G J} or {(xi,..., xn) G Lmkl | xj = 1, V« G In \ J} for any J C In of order l. Proof. A finite subset X of Lmkl satisfies D(X) < D(Lmkl) if and only if {i | xj = — 1,0} U {i | yj = —1,0} is not empty for any distinct (xi,..., xn), (yi,..., yn) G X. We can therefore apply the Erdos-Ko-Rado Theorem [8] to determine the positions of entries —1 or 0. The number of possible positions of —1, 0 is (mi+—- J. After fixing the position, —1, 0 can be placed in (m++k) ways. This determines Mmkl. The largest sets are classified from the optimal sets of the Erdos-Ko-Rado Theorem. □ The remaining part of this section is devoted to proving Mi k2 = Mk = (k + 3)+2, and determining the classification of the largest sets. Note that D(Lik2) = %/10 and if X C L ik2 satisfies D(X) < D(Lik2), then D(X) < a/8. The following two lemmas are used later. Lemma 2.3. Let X C L ik2 with D(X) < D(Lik2). Suppose k > 4, and |X| > Mk. Then there exists i G {1,..., n} such that nj(X, 0) > Mk-i. Proof. This lemma is immediate because the average of nj(X, 0) is n t nj (X, 0) = kX > k+3 = Mk- i — > Mk- i — 1. □ n^ k + 3 k + 3 k + 3 i= i Lemma 2.4. Let G = (V, E) be a connected simple graph, and E' a matching in G. Assume that G has an independent set I of size |V| — |E'|. Then for z G I if x G V satisfies (x, y) G E' for some y adjacent to z, then x G I. 4 Ars Math. Contemp. 13 (2017) 15-21 Proof. Since the cardinality of I is | V| — |E'|, only one of x or y is an element of I for any (x, y) G E'. By assumption, y G I, and hence x G I. □ The subsets Sk(i), Tk(i), Uk(i) of Lik2 are defined by Sk(i) = {(xi, ...,x„) G Lifc2 | xi = • • • = xi-i = 0,xi = —1}, Tk(i) = {(xi, ...,x„) G Lifc2 | xi = • • • = xj-i = 0,xi = 1}, Uk(i)^(xi,---,xn)gLik21 x;={2,xi.=i—,jG={/+i,...,n}} for i = 2, ...,k + 2. We define Sfc (1) = Ni(Lik2, — 1), and Tfc (1) = Ni(Lik2,1). The following are candidates of the largest subsets avoiding the largest distance %/10. k+i Xfc = Tk(k +1) u ( U Sk(i)) for k > 1, i=i k-i Yi = Ti(1), Yk = Tk(k) U ( U Sk(i)) for k > 2, i=i k-2 Z2 = T2(1), Zk = Tk(k — 1) U ( U Sk(i)) for k > 3. i=i Note that |Xk | = | Yk | = |Zk | = Mk, and they can be inductively constructed by Xk = {(0,x) | x G Xk-i} U Ni(Lik2, —1), Yk = {(0,x) | x G Yk-i}U Ni(Lik2, —1), Zk = {(0,x) | x G Zk-i} U Ni(Lik2, —1). We also use the following notation. Xk = Xk \ Sk(1) = {(0, x) | x G Xk-i} (k > 2), Yk' = Yk \ Sk(1) = {(0, x) | x G Yk-i} (k > 2), Zk = Zk \ Sk(1) = {(0, x) | x G Zk-i} (k > 3). Theorem 2.5. Let X c Lik2 with D(X) < D(Lik2). Then we have |X| < Mk. If equality holds, then (1) for k = 1, X = Xi, or Yi, (2) for k > 2, X = Xk, Yk, or Zk, up to isomorphism. This theorem will be proved by induction. We first prove the inductive step. Lemma 2.6. Let k > 2. Assume that the statement in Theorem 2.5 holds for some k — 1. Let X c Lik2 with D(X) < D(Lik2), such that nj(X, 0) = Mk-i for some i. Then we have |X | < Mk. If equality holds, then X = Xk, Yk, or Zk, up to isomorphism. S. Adachi and H. Nozaki: On the largest subsets avoiding the diameter of (0, ±1)-vectors 5 Proof. Without loss of generality, n(X, 0) = Mk-i, and hence X contains X'k, Yk', or Z' for k > 3, and X1, or Y' for k = 2. (i) Suppose X' C X for k > 2. The set of other candidates of elements of X is Sk (1) U Uk (k). The diameter graph G of Sk (1) U Uk (k) is a bipartite graph of the partite sets Sk(1) and Uk(k). Since the three elements (-1,0,..., 0, 0,1,1), (-1,0,..., 0,1,0,1), (-1, 0,..., 0,1,1,0) € Sk (1) are isolated vertices in G, they may be contained in X. Let G' be the subgraph of G formed by removing the three isolated vertices. A perfect matching of G' is given as follows. Matching (i) Sk (1) (-1,x2,...,xk+3) Uk(k) (1,y2,...,yk+3) xi = 1, xj = 1 (2 < i < k, i < j < n) xj = 1, xn = 1 (2 < i < k) yi = -1,yj+i = 1 y = -1,yi+i = 1 By this matching, we can show |X| < Mk_ i + |Sk(1)| = Mk. We will classify the sets attaining this bound. First assume that x € X for some x € Sk(1) with x2 = 1. By Lemma 2.4, X must contain any x € Sk(1) with x2 = 1. In particular, (-1,1,1,0,..., 0) € X. Using Lemma 2.4 again, X must contain x € Sk(1) with x3 = 1. By a similar manner, X must contain any x € Sk(1). Therefore X = Xk. Assume X does not contain any x € Sk (1) with x2 = 1, namely n2(X, 1) = 0. By assumption, we have |X| = n2(X,-1) + n2(X,0) < ^ + ^ + Mk_i = Mk. If |X| = Mk, then we have n2(X, -1) = (k+2) and n2(X, 0) = Mk_ 1. This implies that X is isomorphic to Xk, Yk, or Zk. (ii) Suppose Yk' C X for k > 2. The set of other candidates of elements of X is the union of Sk (1) , Uk ( k - 1) , and Si = {(xi,... ,xk+3) € Lik2 | xi = 1,xk = 1,xj = -1, k < j} for k > 3, and S2(1) U Si for k = 2. The diameter graph G of Sk (1) U Uk (k - 1) U Si is a bipartite graph of the partite sets Sk(1) and Uk (k - 1) U Si. Since the three elements (-1, 0,..., 0,1,1,0,0), (-1,0,..., 0,1, 0,1,0), (-1,0,..., 0,1, 0, 0,1) € Sk (1) are isolated vertices in G, they may be contained in X. Let G' be the subgraph of G formed by removing the three isolated vertices. A perfect matching of G' is given as follows. Matching (ii) Sk (1) (-1,x2,...,xk+3) Uk(k - 1) (1,y2,... ,yk+3) xi = 1, xj = 1 (2 < i < k - 1, i < j < n) xi = 1, xn = 1 (2 < i < k - 1) yi = -1,yj+i = 1 yi = -1,yi+i = 1 6 Ars Math. Contemp. 13 (2017) 15-21 Sk(1) Si (-1,0,..., 0,1,1, 0) (-1,0,..., 0,0,1,1) (-1,0,..., 0,1,0,1) (1, 0,..., 0,1,-1, 0, 0) (1,0,..., 0,1,0,-1,0) (1,0,..., 0,1,0,0,-1) By this maching, we can show |X | < Mk. We will classify the sets attaining this bound. For k = 2, the maximum indepdent sets of G' is {(-1,0,0,1,1), (-1,0,1,0,1), (-1,0,1,1,0)} C S2(1) or Si. This implies that X = Y2 or Z2. For k > 3, we assume that x G X for some x G Sk(1) with x2 = 1. By Lemma 2.4, X must contain any x G Sk (1). Therefore X = Yk. If X does not contain any x G Sk (1) with x2 = 1, namely n2(X, 1) = 0. It can be proved that X is isomorphic to Xk, Yk, or Zk. (iii) Suppose k > 3, and Zk C X. The set of other candidates of elements of X is the union of Sk(1), Uk(k - 2), and S2 = {(xi,... ,xk+3) G Lik2 | xi = 1,xk-i = 1,xj = -1, k < j} for k > 4, and S3(1) U S2 for k = 3. The diameter graph G of Sk(1) U Uk(k - 2) U S2 is a bipartite graph of the partite sets Sk(1) and Uk(k - 2) U S2. Since the four vectors (-1,0,..., 0,1,1, 0, 0,0), (-1,0,..., 0,1,0,1, 0,0), (-1,0,..., 0,1,0, 0,1,0), (-1,0,..., 0,1,0, 0, 0,1) G Sk (1) are isolated vertices in G, they may be contained in X. Let G' be the subgraph of G formed by removing the four isolated vertices. A maximum matching of G' is given as follows. Matching (iii) Sk (1) (-1,x2,...,xk+3) Uk (k - 2) ... ,yk+3) x, = 1, xj = 1 (2 < i < k - 2, i < j < n) x, = 1,xn = 1 (2 < i < k - 2) y = -1,yj+i = 1 y = -1,yi+i = 1 Sk (1) S2 (-1, 0,..., 0,1,1,0, 0) (-1,0,..., 0,0,1,1,0) (-1,0,..., 0,0,0,1,1) (-1,0,..., 0,1,0,0,1) (1, 0,..., 0,1,-1, 0, 0, 0) (1,0,..., 0,1,0,-1,0,0) (1,0,..., 0,1,0,0,-1,0) (1,0,..., 0,1,0,0,0, -1) Note that the two vectors (-1,0,..., 0,1, 0,1,0), (-1,0,..., 0,0,1, 0,1) G Sk(1) (2.1) are unmatched in this matching. By this matching, we can show |X | < Mk. We will classify the sets attaining this bound. If |X | = Mk, then the two vectors in (2.1) must be contained in X. Therefore X does not contain any element of S2, and contains an element of Sk (1) which matches some element of S2. For k = 3, X therefore contains Sk (1), and X = Z3. For k > 4, we assume that x G X for some x G Sk(1) with x2 = 1. By Lemma 2.4, X must contain any x G Sk(1). Therefore X = Zk. If X does not contain any x G Sk(1) with x2 = 1, namely n2(X, 1) = 0. Therefore X is isomorphic to Xk, Yk, or Zk. □ S. Adachi and H. Nozaki: On the largest subsets avoiding the diameter of (0, ±1)-vectors 7 Matchings (i)-(iii) and the notation Si, S2 defined in the proof of Lemma 2.6 are used again later. The base case in the induction is the case k = 3. We will prove the cases k = 1,2, 3 in order. Proposition 2.7. Let X c L112 with D(X) < D(L112). Then we have |X | < M1 =6. If equality holds, then X = X1; or Y1, up to isomorphism. Proof. Since the diameter graph G of L112 is isomorphic to C4 U C4 U C4, where C4 is the 4-cycle, the bound |X | < 6 clearly holds. Considering the permutation of coordinates, G has the automorphism group S4. Since the stabilizer of X1 in S4 is of order 6, the orbit of X4 has length 4. Similarly the orbit of Y1 has length 4. Since the number of maximum independent sets of G is 23 = 8, this proposition follows. □ For k = 2, we also classify (M2 - 1)-point sets X with D(X) < D(L122) in order to prove the case k = 3. Proposition 2.8. Let X c L122 with D(X) < D(L122). Then we have |X| < M = 12. If |X| = 12, then X = X2, Y2, or Z2, up to isomorphism. If |X| = 11, then X is v2 = X2 u {(-1, o, o, 1,1), (-1, o, 1, o, 1), (-1, o, 1,1,0), (-1,1,1, o, o), (1,-1,1, o, o)}, W2 = Y2' U {(-1,1,1,0, 0), (-1,1, 0,1,0), (-1,1,0, 0,1), (-1, 0,0,1,1), (1,1,-1, 0,0)}, or the set obtained by removing a point from X2, Y2, or Z2, up to isomorphism. Proof. First suppose n4(X, 0) = 6 for some i. Then we have |X| < 12, and X with |X| = 12 is X2, Y2, or Z2 by Lemma 2.6. In order to find X with |X| = 11, we consider 5-point independent sets in the diameter graph of S2(1) U U2(2) or S2(1) U U2(1) U Si. If X is not isomorphic to a subset of X2, Y2, or Z2, then X = V2 from S2(1) U U2(2), and X = W2 from S2(1) U U2(1) U S1. Suppose ni(X, 0) < 5 for any i. If |X| > 11, then the average of n4(X, 0) is greater than 4. Without loss of generality, we may assume h1(X, 0) = 5. Since the diameter graph of L112 is C4 U C4 U C4, we can show that X contains a 5-point subset of X2 or Y2. (i) Suppose X contains a 5-point subset of X2. By considering the automorphism group of X2, we may assume X contains the 5-point subset obtained by removing (0, -1,0,1,1) or (0,0, -1,1,1). First assume that X contains the 5-point subset obtained by removing (0, -1,0,1,1). Since other candidates of elements of X are still in S2(1) U U2(2), we have |X| < 11, and if |X| = 11, then X is isomorphic to a subset of X2, Y2, or Z2. Assume that X contains the 5-point subset obtained by removing (0,0, -1,1,1). The set of other candidates of elements of X is £2(1) U ^(2) U {(1,0,1, -1,0), (1,0,1,0, -1)}. If X does not contain both (1,0,1, -1,0) and (1,0,1,0, -1), then |X| < 11, and X 8 Ars Math. Contemp. 13 (2017) 15-21 attaining this bound is isomorphic to a subset of X2, Y2, or Z2. To make a new set, X may contain (1,0,1, -1,0). The two vectors (-1,1,0,1,0), (-1,0,0,1,1) G £2(1), which are at distance %/10 from (1,0,1, -1,0), are not contained in X. The set P1 consisting of the two isolated vertices (-1, 0, 1, 0, 1), (-1, 0,1,1, 0) G S2(1) and 6 points (-1,1,1, 0, 0), (-1,1,0,0,1), (1, -1,1, 0,0), (1, -1,0,1, 0), (1,-1,0,0,1), (1,0,1,0, -1) has the unique maximum 6-point independent set I (-1, 0,1,0,1), (-1, 0,1,1,0), (1, -1,1, 0, 0), 1 \ (1,-1, 0,1,0), (1,-1, 0, 0,1), (1, 0,1,-1, 0) which gives X isomorphic to Y2, and n2(X, 0) = 6. If X contains a 5-point independent set in P1 and is not isomorphic to a subset of Y2, then X contains the 5-point independent set {(-1, 0,1,0,1), (-1,0,1,1, 0), (-1,1,1,0,0), (1, -1,1, 0,0), (1, 0,1,0, -1)}. Then X is isomorphic to W2 and n2 (X, 0) = 6. (ii) Suppose X contains a 5-point subset of Y2. By considering the automorphism group of Y2', we may assume X contains the 5-point subset obtained by removing (0,1, -1,0,1). The set of other candidates of elements of X is S2 (1) U S1 U {(1,0,1,0, -1)}. To make a new set, X may contain (1,0,1,0, -1). The two vectors (-1,1,0,0,1), (-1,0,0,1,1) G S2(1), which are at distance %/10 from (1,0,1,0, -1), are not contained in X. The set consisting of the two isolated vertices (-1,1, 1, 0, 0), (-1, 1, 0,1, 0) G S2(1) and 5 points (-1,0,1,1,0), (-1, 0,1,0,1), (1,1, -1,0,0), (1,1,0, -1, 0), (1,1, 0, 0, -1) has the unique maximum 5-point independent set {(-1,1,1,0,0), (-1,1,0,1, 0), (1,1, -1,0,0), (1,1,0,-1,0), (1,1, 0,0,-1)}, which gives X is isomorphic to a subset of Z2. □ Proposition 2.9. Let X c L132 with D(X) < D(L132). Then we have |X| < M3 = 22. If equality holds, then X = X3, Y3, or Z3, up to isomorphism. Proof. If n(X, 0) = 12 for some i, then we have |X| < 22, and the set attaining this bound is X3, Y3, or Z3 by Lemma 2.6. S. Adachi and H. Nozaki: On the largest subsets avoiding the diameter of (0, ±1)-vectors 9 Suppose nj(X, 0) < 11 for any i. If |X| > 22, then the average of n4(X, 0) is greater than 11, which gives a contradiction. Therefore |X | < 22, and if |X | = 22, then the average of n (X, 0) is 11, and n4(X, 0) = 11 for any i. By Proposition 2.8, X may contain V3' = {(0, v) e L132 | v e V2}, W = {(0,w) e L132 | w e W2}, or an 11-point set obtained by removing a point from Xg, Y3', or Zg. (i) Suppose X contains an 11-point subset of Xg. By considering the automorphism group of Xg, X may contain the set in Xg obtained by removing (0, -1,0,0,1,1), (0, -1, 1,1,0,0), (0,0, -1,0,1,1), or (0,0,0, -1,1,1). If X contains the set Xg with (0, -1,0,0, 1,1), (0, -1,1,1,0,0), or (0,0, -1,0,1,1) removed, then the set of other candidates of X is still £3(1) U U3(3), and |X| < 22. Suppose X contains the set Xg with (0,0,0, -1,1,1) removed. Then new candidates of vectors of X are only (1,0,0,1, -1,0) and (1,0,0,1,0, -1), and X may contain (1,0,0,1, -1,0). The three vectors (-1,1,0,0,1,0), (-1,0,1, 0,1,0), and (-1,0,0,0,1,1), which are at distance V'W from (1,0,0,1, -1,0), are not contained in X. Therefore by |X | = 22, the other new candidate (1,0,0,1,0, -1), and two isolated vectors (-1,0,0,1,0,1), and (-1,0,0,1,1,0) must be contained in X. Moreover a 7-point independent set must be obtained from Matching (i). Since (-1,1,0,0,1,0) and (-1,0,1,0,1,0) are not contained in X, by Lemma 2.4, (1, -1,0,0,0,1) and (1,0, -1,0, 0,1) must be contained in X, and consequently any element of U2(2) is contained in X. This implies n2(X, 1) = 0, and X is isomorphic to X3, Y3, or Z3. (ii) Suppose X contains an 11-point subset of Y3'. By considering the automorphism group of Y3', X may contain the set in Y3' obtained by removing (0, -1,0,0,1,1), (0, -1,1, 1,0,0), or (0,0,1, -1,0,1). If X contains the set Y3' with (0, -1,0,0,1,1), or (0, -1,1,1, 0,0) removed, then the set of other candidates of X is still S3(1) U U3(2) U S1, and |X | < 22. Suppose X contains the set Y3' with (0,0,1,-1,0,1) removed. Then a new candidate of an element of X is only (1,0,0,1,0, -1), and X may contain (1,0,0,1,0, -1). The three vectors (-1,1,0,0,0,1), (-1,0,1,0,0,1), and (-1,0,0,0,1,1), which are at distance %/10 from (1,0,0,1,0, -1), are not contained in X. By considering Matching (ii), we can show |X | < 22. (iii) Suppose X contains an 11-point subset of Z3. By considering the automorphism group of Z3, X may contain the set in Z3 obtained by removing (0,1, -1,0,0,1). Then a new candidate of an element of X is only (1,0,1,0,0, -1), and X may contain (1,0,1,0,0, -1). The three vectors (-1,1,0,0,0,1), (-1,0,0,1,0,1), and (-1,0,0,0,1,1), which are at distance %/10 from (1,0,1,0,0, -1),are not contained in X .By considering Matching (iii), we can show | X| < 22. (iv) Suppose X contains V3'. The set of other candidates of X is S3(1) U U3(3) \ {(1, -1,1,0,0,0)}, and the maximum independent set is of order at most 10 by Matching (i). Thus |X | < 22. (v) Suppose X contains W3. The set of other candidates of X is S3(1) U U3(2) US1 \ {(1, -1,0,1,0,0)}, and the maximum independent set is of order at most 10 by Matching (ii). Thus |X| < 22. Therefore this proposition follows. □ Finally we prove Theorem 2.5. 10 Ars Math. Contemp. 13 (2017) 15-21 Proof of Theorem 2.5. By Propositions 2.7-2.9, the statement holds for k = 1,2, 3. By the inductive hypothesis and Lemma 2.3, if |X| > Mk, then there exists i e {1,..., n} such that n4(X, 0) = Mk-1 for k > 4. By Lemma 2.6, this theorem holds for any k. □ 3 Classification of the largest 4-distance sets which contain J(n, 4) A finite set X in Rd is called an s-distance set if the set of Euclidean distances of two distinct vectors in X has size s. The Johnson graph J(n, m) = (V, E), where V = {{ii,..., im} | 1 < ii < • • • < im < n, ij e Z}, E = {(v, u) | |v O u| = m — 1, v, u e V}, is represented into Rn-1 as the m-distance set J(n, m) = L0,n-m,m. Indeed J(n, m) c Rn, but the summation of all entries of any x e J(n, m) is m, and J(n, m) is on a hyperplane isometric to Rn-1. Bannai, Sato, and Shigezumi [1] investigated m-distance sets containing J(n, m). In their paper, for m < 5 and any n, the largest m-distance sets containing J(n, m) are classified except for (n, m) = (9,4). In this section, the case (n, m) = (9,4) will be classified. The set of Euclidean distances of two distinct points of J(9,4) is {v^, V4, >/6, v^}. The set of vectors which can be added to J(9,4) while maintaining 4-distance is the union of the following sets [1]. X= X(iii) = X(ii) = X= 4 where the exponents inside indicate the number of occurrences of the corresponding numbers, and the exponent P outside indicates that we should take every permutation. They conjectured that J(9,4) U X(i) U X(iii) U {(—4/3, (2/3)8)} U X(iv)' is largest, where (—4/3, (2/3)8) e X(ii), and X(iv)' = j(xi,...,xg) e X(i ) I xi 3 4 3 ' 4 , 3, 2 4 3, 3 Actually X(iv)' is isometric to Xe in Section 2 by replacing —2/3, 1/3, 4/3 to —1, 0, 1, respectively. Let X(resp. ) be the set obtained from Ye (resp. Ze) by the same manner. Using Theorem 2.5, we can classify the largest 4-distance sets containing J(9,4). Theorem 3.1. Let X c {(x1,...,x9) which contains J(9,4). Then we have | x1 + • • • + xg = 1} be a 4-distance set |X| < 258. p 8 3 2 e 1 2 3 3 e 2 4 2 u 3 3 t If equality holds, then X is one of the following, up to permutations of coordinates. S. Adachi and H. Nozaki: On the largest subsets avoiding the diameter of (0, ±1)-vectors 11 (1) J(9,4) U X(i) U X(iii) U {(-4/3, (2/3)8)} U X(iv)', (2) J(9,4) U X(i) U X(iii) U {(-4/3, (2/3)8)} U X(iv)'', (3) ,7(9,4) U X(i) U X(iii) U {(-4/3, (2/3)8)} U X(iv)'''. Proof. For any x G X(i) U X(iii), y G u4=1X(j), the Euclidean distance of x, y is i in {a/2, v4, v6, v8}, and hence X may contain X(i) U X(iii). The set X(iv) is isometric to L162 by replacing -2/3, 1/3, 4/3 to -1, 0, 1, respectively. Therefore the largest subsets of X(iv) with distances {V2 V4 V6, V8} are X(iv)', X(iv)'', and X(iv)''', up to permutations of coordinates. If X does not contain any element of X(ii), then |X | < |J(9, 4) U X(i) U X (iii)| + |X(iv)' | = 257. If X contains x G X(ii) with xj = -4/3, then X cannot contain y G X(iv) with yi = 4/3. By re-ordering the vectors, we may assume that the set X(ii) (t) = {x G X(ii) | xj = -4/3, 3i G {1,..., t}} is in X for some t. Clearly, from the definition of X (jj)(t), this set must have size t. For t = 7,8, 9, X contains at most one element of X(iv), and hence |X | < |J(9,4) U X(i) U X (iii)| + t + 1 < 181. If the set X(ii)(t) is in X for 1 < t < 6, then consider the set of vectors in X n X(iv) in which the entry 1/3 occurs in all of the first t positions. The final 9 - t entries of one of these vectors forms a vector from L1,6-t,2; no two vectors in this set can be at the maximum distance. Thus the size of |{x G X n X(iv) | xj = 1/3, Vi G {1, .. ., t}}| is bounded by M6-t. It is clear that |{x G X n X(iv) | xj = -2/3, xjl = 4/3, xj2 = 4/3, 3i G {1,...,t}, 3ji,j2 G{t +1,..., 9}}| is bounded by t (9-1). Thus, for 1 < t < 6, we have |X| < |J(9,4) U X(i) U X(iii) | +1 + M6-t + ^9 - ^ t3 9t2 31t =----1---+ 257 < 258, 3 2 6 < ' and equality holds only if t = 1. The sets attaining this bound are only the three sets in the statement. □ 4 Remarks on other Mmkl Actually it is hard to determine Mmkl for other (m, k, l) by a similar manner in Section 2. Fix m, l, where m < l. By Proposition 2.2, if k < l - m, then Mmkl = (mr—-1)(m"lk). 12 Ars Math. Contemp. 13 (2017) 15-21 In general there are many largest sets for k = l — m. For k > l — m, we can inductively construct a large set Xk C Lmkl satisfying D(Xk) < D(Lmkl) as follows Xk = {(0, x') | x' € Xfc_i} U {(xi,. .., x„) € Lmkl | xi = —1}, where Xl-m is a largest set for k = l — m. Therefore we have _ fm + l- 1N fk + m + A /m + l-T Mmkl > Mmki := + 1 + , + + + ym — 1 J \ m + l J \ m We can generalize Lemma 2.3 as follows. Lemma 4.1. Let X C Lmkl with D(X) < D(Lmkl). Suppose k > m(— m — l +1, and |X| > Mmkl. Then there exists i € {1,..., n} such that Ui(X, 0) > Mm,k-1,l. Proof. This lemma is immediate because the average of ni (X, 0) is 1 £ ni(X, 0)= k|X| > kMmkl n m + k + l m + k + l i=1 m +1 fm + k + A _ = Mm,k-1,l--^^ , > Mk-1 — 1. □ m + k + l l In the manner of Section 2, it is hard to classify Mmkl for m — l + 1 < k < m (— m — l. Moreover it seems to be difficult to give matchings, like Matching (i) or (ii), of many possibilities of Xk. We need another idea to determine other Mmkl. 5 Acknowledgments The authors thank Sho Suda for providing useful information. References [1] E. Bannai, T. Sato and J. Shigezumi, Maximal m-distance sets containing the representation of the Johnson graph J(n, m), Discrete Math. 312 (2012), 3283-3292, doi:10.1016/j.disc.2012.07. 028, http://dx.doi.org/10.1016/j.disc.2012.07.028. [2] P. Borg, Intersecting families of sets and permutations: a survey, Int. J. Math. Game Theory Algebra 21 (2012), 543-559 (2013). [3] P. Delsarte, J. M. Goethals and J. J. Seidel, Spherical codes and designs, Geometriae Dedicata 6 (1977), 363-388. [4] M. Deza and P. Frankl, Every large set of equidistant (0, +1, —1)-vectors forms a sunflower, Combinatorica 1 (1981), 225-231, doi:10.1007/BF02579328, http://dx.doi.org/10. 1007/BF02579328. [5] M. Deza and P. Frankl, Erd6s-Ko-Rado theorem—22 years later, SIAM J. Algebraic Discrete Methods 4 (1983), 419-431, doi:10.1137/0604042, http://dx.doi.org/10.1137/ 0604042. [6] M. Deza and P. Frankl, On t-distance sets of (0, ±1)-vectors, Geom. Dedicata 14 (1983), 293301, doi:10.1007/BF00146909, http://dx.doi.org/10.100 7/BF0 014 690 9. [7] M. Deza and P. Frankl, Bounds on the maximum number of vectors with given scalar products, Proc. Amer. Math Soc. 95 (1985), 323-329, doi:10.2307/2044537, http://dx.doi.org/ 10.2307/2044537 . S. Adachi and H. Nozaki: On the largest subsets avoiding the diameter of (0, ±1)-vectors 13 [8] P. Erdos, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser. (2) 12 (1961), 313-320. [9] P. Frankl, The shifting technique in extremal set theory, in: Surveys in combinatorics 1987 (New Cross, 1987), Cambridge Univ. Press, Cambridge, volume 123 of London Math. Soc. Lecture Note Ser., pp. 81-110, 1987. ¿^creative , ars mathematica ^commons contemporánea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 15-21 The distinguishing index of the Cartesian product of countable graphs Izak Broere * Department of Mathematics and Applied Mathematics, University of Pretoria Monika Pilsniak t AGH University, Department of Discrete Mathematics, Krakow, Poland Received 6 January 2015, accepted 12 July 2016, published online 11 August 2016, corrected 12 January 2017 Abstract The distinguishing index D'(G) of a graph G is the least cardinal d such that G has an edge colouring with d colours that is preserved only by the trivial automorphism. We derive some bounds for this parameter for infinite graphs. In particular, we investigate the distinguishing index of the Cartesian product of countable graphs. Finally, we prove that D'(k2^° ) = 2, where K2N° is the infinite dimensional hypercube. Keywords: Distinguishing index, automorphism, infinite graph, edge colouring, infinite dimensional hypercube. Math. Subj. Class.: 05C25, 05C80, 03E10 1 Introduction Albertson and Collins [1] introduced the (vertex-)distinguishing number D(G) of a graph G as the least cardinal d such that G has a labelling with d labels that is only preserved by the trivial automorphism. This concept has spawned numerous papers, mostly on finite graphs. But countable infinite graphs have also been investigated with respect to the distinguishing number; see [12], [13], and [14]. For graphs of higher cardinality, see [8]. The corresponding notion for endomorphisms instead of automorphisms is investigated in [5]. Let us consider now any edge colouring of a graph G; it is merely a function f : E(G) ^ C which labels each edge of G with a colour from some set C. Given a graph *This author is thankful to the AGH University of Science and Technology whose hospitality he enjoyed during the preparation of this paper; he is also supported in part by the National Research Foundation of South Africa (Grant Numbers 90841, 91128). tThe research was partially supported by the Polish Ministry of Science and Higher Education. E-mail addresses: izak.broere@up.ac.za (Izak Broere), pilsniak@agh.edu.pl (Monika Pilsniak) ©® This work is licensed under http://creativecommons.Org/licenses/by/3.0/ 16 Ars Math. Contemp. 13 (2017) 15-21 G with an edge colouring f, we say that a graph automorphism < : V(G) ^ V(G) of G preserves the edge colouring f if f (xy) = f (<(x)<(y)) for every edge xy e E(G); if, on the other hand, there is an edge xy such that f (xy) = f (<(x)<(y)), then we say that < is broken by xy. It is easy to see that there is, for every connected graph G = K2, an edge colouring of G which is preserved only by the trivial automorphism of G, i.e., only by the identity idG : V(G) ^ V(G): Merely choose different colours for different edges. The distinguishing index D'(G) of a graph G is the least cardinal d such that G has an edge colouring with d colours that is only preserved by the trivial automorphism. Obviously for K2 the distinguishing index is not defined and it is the only such connected graph. For finite graphs this concept is investigated by Kalinowski and Pilsniak in [9] and by Pilsniak in [11]. In [2], the following general upper bound was proved. Theorem 1.1. Let G be a connected, infinite graph such that the degree of every vertex of G is not greater than A. Then D'(G) < A. A graph G is said to be prime with respect to the Cartesian product if whenever G = G1mG2, then either Gi or G2 is the graph consisting of a single vertex. It is well known (see [6]) that if G is connected, then G has a unique prime factorization, i.e., G = G1DG2D ■■■ OGt such that for 1 < i < t, Gi is prime. Two graphs G and H are called relatively prime if K1 is the only common factor of G and H. About forty-five years ago Imrich and Miller independently proved the following theorem - see Thm. 6.10 in [6]. Theorem 1.2. If G is connected and G = G1DG2D ■ ■ ■ OGr is its prime decomposition, then every automorphism of G is generated by the automorphisms of the factors and the transpositions of isomorphic factors. A basic fact, which is a reformulation of the above theorem for r = 2 and which is used frequently in this paper, is: If < is an automorphism of the Cartesian product G1mG2 of two connected relatively prime graphs, then there are automorphisms 2. Then D'(Gk) = 2 with the only exception: D'(K2) = 3. 2 The distinguishing index of the Cartesian product First we consider the Cartesian product of two denumerable graphs with infinite edge sets. Lemma 2.1. Let Gi and G2 be two connected relatively prime denumerable graphs. Then D'(GiDG2) < 2. Proof. We start by labelling the edges of Gi with ei, e2,... and those of G2 with fi, f2,... This is possible since both edge sets have to be denumerable. Note that these labellings effectively order the edges of these graphs. We can now easily describe the required edge distinguishing colouring in colours 1 and 2: Colour the first (in terms of the above ordering) k edges of the k'th layer of Gi and the first k edges of the k'th layer of G2 with 1; colour all other edges with 2. Recall that every edge in GiDG2 lies in a Gi-layer or a G2-layer; hence this process colours indeed all edges of GinG2. Using the labels, this means that the edges corresponding to the edges {ei, e2,..., ek} of Gi in the k'th Gi-layer and the edges corresponding to the edges {fi, f2,..., fk} of G2 in the k'th G2-layer, for all k = 1, 2,..., are coloured 1 and all other edges are coloured 2. Now consider, if possible, any non-trivial automorphism ^ = (^i,^2) of GiDG2 which preserves the above edge colouring of Gi mG2. Since every two different Gi-layers have different numbers of edges coloured with 1, the automorphism ^>2 of G2 must be trivial. Similarly, must be trivial. Hence ^ is the trivial automorphism, proving that for every non-trivial automorphism ^ of Gi DG2 there is an edge e of Gi DG2 for which e and y(e) are coloured differently. □ The same result was obtained for the distinguishing number of two connected relatively prime denumerable graphs by Imrich and Klavzar in [7]. Recently it was shown by Estaji, Imrich, Kalinowski, Pilsniak and Tucker in [3] that the condition that the two graphs are relatively prime can be omitted. Note that Lemma 2.1 assures us that D'(GinG2) is at most two irrespective of the values of D'(Gi) and D'(G2). Next we consider the case in which both Gi and G2 of orders being any cardinals and with finite values for the distinguishing index. Lemma 2.2. Suppose Gi and G2 are connected relatively prime graphs with finite distinguishing indexes. If D'(Gj) < k, i = 1,2, then D'(GiDG2) < max{ki, k2}. Proof. Since D'(Gj) < kj, i = 1,2, there are, with k = max{ki, k2}, edge colourings fi of Gi and f2 of G2 using the colours 1,2,..., k which are distinguishing colourings of Gi and G2 respectively. In order to prove now that D'(Gi DG2) < k, we again use the notion of a "first" layer through a labelling of the vertices (which here is not explicitly chosen or named). Hence consider the function f : E(GiDG2) ^ {1, 2,..., k} defined by 1) f ((vi, w)(v2, w)) = fi(viv2) for edges of the first Gi-layer and 18 Ars Math. Contemp. 13 (2017) 15-21 2) f ((v, wi)(v, w2)) = /2(wiw2) for edges of the first G2-layer and 3) f (e) = 1 for all remaining edges. Consider any non-trivial automorphism a = (a1, a2) of G1mG2 with a1 a non-trivial automorphism of G1 or a2 a non-trivial automorphism of G2. Assume that the first is true (for G1): Then, since f1 is a distinguishing colouring of the first G1-layer, there is an edge e of G1 such that f1(e) = f1(a1(e)). Now, if a2 does not move the first layer, then this edge (considered as an edge of G1 mG2) is an edge of the required kind in the first G1-layer. On the other hand, if a2 does move the first layer to another layer, we can remark, since f1(e) = f1(a1(e)), that at least one of f1(e) and f1(a1(e)) is different from 1 so that this edge is moved by a2 to an edge in another layer which has colour 1 by 3) above. A similarly argument holds if the second is true (for G2) - merely interchange the roles of G1 and G2 (and their colourings and automorphisms) in the above argument. Hence we are assured that all non-trivial automorphisms of G1mG2 are broken by the colouring f. □ Observe, that D'(G1DG2) can be arbitrary large, for instance if G1 is isomorphic to and G2 is isomorphic to an infinite ray with many (but finitely many) leaves adjacent to its first vertex. In our next result we prove that if G1 satisfies D'(G1) = H0 and the graph G2 is finite (so that, in particular D'(G2) is finite), then D'(G1DG2) = H0. Lemma 2.3. Suppose G1 and G2 are connected relatively prime graphs with D'(G1) = H0 and G2 is finite. Then D' (G1DG2) = K0. Proof. Suppose, for a proof by contradiction, that D'(G1DG2) is finite. Since G2 is a finite graph, there are finite values for ||G2||, the number of edges of G2, and D'(G2) too. Hence we can choose a positive integer k such that each of these three numbers is at most k. Since D'(G1DG2) < k, there is a k-distinguishing edge colouring f of the edges of G1DG2. Furthermore, since D'(G1) = H0, there exists, for every positive integer t, anon-trivial automorphism at of G1 which needs at least t + 1 colours to break it. So if t > k, the colouring by f of any layer of G1 induces a colouring on G1 which cannot be broken by the automorphism at of G1. Since there are infinitely many such automorphisms, we may assume without loss of generality that as = at when s = t. Now consider non-trivial automorphisms of G1DG2 of the form a = (at, idG2) (for some t > k). For each such t, and each edge vw of G1 (which we can consider as an edge of any G1-layer of G1DG2), we have that f (vw) = f (at(v)at(w)), i.e., these automorphisms of G1nG2 are not broken by edges in layers of G1. The automorphisms a of the above form should therefore be broken by edges of layers of G2. But this means that, for each t > k, for at least one edge xy of the G2-layer determined by a vertex v g V(G1), we have that f (xy) in this layer is different from f (at(x)at(y)) in the G2-layer determined by at(v) g V(G1). Since there are infinitely many G2-layers, this requires infinitely many different colourings of G2. However, there are at most k||G2 ^different colourings of G2-layers. Hence the colouring f cannot break all the infinitely many automorphisms described above. □ As a consequence of the above three lemmas we immediately obtain the following characterisation. I. Broere andM. Pilsniak: The distinguishing index of the Cartesian product. 19 Theorem 2.4. If Gi and G2 are connected relatively prime countable graphs, then D'(GiDG2) is infinite if and only if for some i G {1,2} we have that D'(Gj) is infinite while for j = i we have that Gj is finite. □ Now we consider a graph which is the Cartesian power Gk of a denumerable graph G. For a finite graph G, the distinguishing number of the Cartesian power of G is considered in [4]. Here we prove a result for graphs G with a finite number of prime factors (counted with their multiplicities). We begin with a result for prime graphs. Lemma 2.5. Let k > 2 be an integer. If a connected denumerable graph G is prime with respect to the Cartesian product, then D' (Gk) = 2. Proof. If k = 2, the proof is similar to the proof of Lemma 2.1. Indeed, denote G2 = GiDG2, where Gi, G2 are isomorphic to G. Using an analogous proof technique but colouring distinct even numbers of edges of each Gi -layer with red and distinct odd numbers of edges of each G2 -layer with red will also take care of the additional automorphisms generated by the isomorphism between Gi and G2. Now we show that D'(GDH) = 2 if D'(H) = 2 and G is prime. In particular, if we consider H = Gk-i then we obtain the thesis by induction. Namely, let f be a distinguishing colouring of H with two colours. We define a colouring of GdH as follows: One H-layer is given the colouring f, hence all automorphisms of this H-layer are broken. We colour another H-layer completely blue and all remaining H-layers we colour with distinct numbers of red edges different from the number of red edges in f. Hence all automorphisms of G are broken. If G' isomorphic with G is a factor of H, then we have additional automorphisms, generated by interchanging of G and G'. To break them, we colour each G-layer red. Then every G'-layer contained in a blue H-layer is completely blue, so it cannot be interchanged with G. In this way we break all nontrivial automorphisms of GDH with two colours if D'(H) = 2 and G is prime. □ The above proof is analogous to the proof of a similar result in [7]. Observe that D'(GDH) = 2 if D'(H) = 2 and G is prime, also if G is finite. Theorem 2.6. Let k > 2 be an integer and G be a connected denumerable graph with the prime factor decomposition G = GiD...DGr, where Gi,..., Gr are not necessarily distinct. Then D'(Gk) = 2. Proof. If G is prime, the claim follows from Lemma 2.5. If G is not prime, we consider the prime factorization G = GiD...DGr and apply Lemma 2.5 to every infinite factor (G has at least one infinite prime factor). Moreover, we can use Theorem 1.4 for every finite factor. The result then follows from Lemma 2.2 unless G = K2DH and k = 2, where H is an infinite graph relatively prime with K2. But we already know that D'(H2) = 2 due to the above arguments, so let f be a distinguishing colouring of H2 with two colours. We then define a colouring of G2 in terms of its four H2-layers as follows: One H2-layer is given the colouring f, hence all automorphisms of this H2-layer are broken. The three remaining H2-layers are coloured with distinct numbers of red edges (while all remaining edges are blue), hence all automorphisms of G2 are broken. □ We say the G has infinite diameter if there are vertices of arbitrarily large distance. Such a situation occurs in particular in any weak Cartesian product G of infinitely many non-trivial factors (finite or infinite). Hence the above theorem immediately implies the following. 20 Ars Math. Contemp. 13 (2017) 15-21 Corollary 2.7. Let k > 2 be an integer and let G be a connected denumerable graph with finite diameter. Then D'(Gfc) = 2. 3 The distinguishing index of the infinite hypercube The situation is quite different when we have infinitely many factors in the Cartesian power - consider for example the infinite dimensional hypercube K^0. This (uncountable) graph has vertices represented by (denumerable) sequences of 0's and 1's and two vertices are adjacent whenever their binary sequences differ in exactly one entry. This graph also has uncountably many connected components, each a countable graph, which are pairwise isomorphic. The automorphism group of K^'0 is well described (see [10]). Using this information, we are now ready to prove Theorem 3.1. Let K^0 be the infinite dimensional hypercube. Then 0) = 2. Proof. We first construct an asymmetric spanning tree and then show how it can be used to prove the existence of an asymmetric spanning subgraph in every component of K^0; these subgraphs will be constructed in such a way that different components have non-isomorphic subgraphs. Towards the end of the proof, we shall show how they can be exploited to break all non-trivial automorphisms of the hypercube K^00. It is convenient to describe the required asymmetric subgraphs by first handling the connected component C0 in which all sequences have only finitely many 1's (and therefore an infinite tail of 0's). First we build an asymmetric tree T, which is a spanning subgraph of C0, as follows: Take (0,0,0,0,...) and let it be the central vertex. Then add (1,0,0,0,...), and the edge between it and the central vertex, to form the first branch of the tree. Next take (0,1,0,0,0,...) and (1,1,0,0,0,...) and the path between them and the central vertex to form the second branch of the tree. The i'th branch of this tree will therefore be the path on the central vertex and (0i-1,1,0,0,0,...), (0i-2,1,1,0,0,0,...),... and will have length 2i-1. All these binary sequences have 1 on the i'th entry and if we restricted them to the first i - 1 entries, then we obtain the binary-reflected Gray code list with i - 1 bits. It can be generated recursively from the list for i - 2 bits by reflecting the list (i.e. listing the entries in reverse order), concatenating the original list with the reversed list, prefixing the entries in the original list with 0, and then prefixing the entries in the reflected list with 1. In particular, the last vertex of the i'th branch has the code (1,0i-2,1,0,0,0,...), and the last but one has the code (1,0i-3,1,1,0,0,0,...). Note that all branches of T are of different length, which ensures us that T is asymmetric, and note that it is a spanning tree of the component C0. So it means that we can easily distinguish the weak Cartesian product of H0 copies of K2 by two colors: Namely we colour all the edges of T with one colour and the remaining edges with the second colour. Now we would like to distinguish the Cartesian product of K0 copies of K2 by two colours. Consider any sequence x = (x1, x2,...) of 0's and 1's and suppose it is in the connected component C of the hypercube K20. Since C is isomorphic to C0, we can find a copy of T, say TC, in C. Now we use x and TC to create a spanning subgraph of C by adding edges to TC as follows: For every positive integer i we add the edge of K20 between the endvertex of the i'th branch and the last but one vertex of the (i + 1)'th branch of TC to this tree if and only if I. Broere andM. Pilsniak: The distinguishing index of the Cartesian product. 21 xi = 1. We remark that this edge is indeed in K20 since the binary sequences representing these vertices in C0 differ in exactly one entry, namely the (i + 1)'th entry, and therefore the same is true in the isomorphic copy TC of T. Note also that the choice of the added edges ensures us that Tx is not isomorphic to Tx> whenever x = x'. Since there are uncountably many sequences x, we thus have uncountably many pairwise non-isomorphic subgraphs all of which are asymmetric. Finally we prove, using these subgraphs of the components of K2, that the infinite hypercube is 2-distinguishable. Consider the following colouring f of the edges of K^0: Colour, for each component C of K20 and some fixed choice of a vertex x of C, all the edges of the spanning subgraph TX with 1; colour all the other edges of K20 with 2. Then consider any automorphism a of K2°. Since isomorphisms, and thus a, preserve connectivity, a has to take every component C of K2 to a component C' of K2. But, if C = C', then the asymmetric spanning subgraphs T£ and T% of C and C' are not isomorphic (because x = x'), hence the colouring f breaks a. □ References [1] M. O. Albertson and K. L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996), R18. [2] I. Broere and M. Pilsniak, The distinguishing index of some infinite graphs, Electron. J. Combin. 23(1) (2015), P1.78. [3] E. Estaji, W. Imrich, R. Kalinowski, M. Pilsniak and T. Tucker, Distinguishing number for the Cartesian product of countable graphs, Discuss. Mathem. Graph Th., http://dx.doi. org/10.7151/dmgt.1902. [4] A. Gorzkowska, R. Kalinowski and M. Pilsniak, Distinguishing the Cartesian product of finite graphs, Ars Math. Contemp. 12 (2017), 77-87. [5] W. Imrich, R. Kalinowski, F. Lehner and M. Pilsniak, Endomorphism breaking in graphs, Electron. J. Combin. 21(1) (2014), P1.16. [6] W. Imrich and S. KlavZar, Product Graphs, John Wiley & Sons, Inc. New York, 2000. [7] W. Imrich and S. KlavZar, Distinguishing Cartesian powers of graphs, J. Graph Theory 53 (2006), 250-260. [8] W. Imrich, S. KlavZar and V. Trofimov, Distinguishing infinite graphs, Electron. J. Combin. 14 (2007), R36. [9] R. Kalinowski and M. Pilsniak, Distinguishing graphs by edge-colourings, Europ. J. Combin. 45 (2015). [10] M. Pankov, A note on automorphisms of the infinite-dimensional hypercube graph, The Electron. J. Comb. 19(4) (2012), P23. [11] M. Pilsniak, Edge Motion and the Distinguishing Index, Preprint MD 076, http://www. ii.uj.edu.pl/preMD. [12] S. M. Smith, T. Tucker and M. E. Watkins, Distinguishability of infinite groups and graphs, Electron. J. Combin. 19 (2012), P27. [13] T. Tucker, Distinguishing maps, Electron. J. Combin. 18 (2011), R50. [14] M. E. Watkins and X. Zhou, Distinguishability of locally finite trees, Electron. J. Combin. 14 (2007), R29. /^creative ^commor ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 23-30 Classification of convex polyhedra by their rotational orbit Euler characteristic Jurij Kovic * IMFM, Jadranska 19, 1000 Ljubljana, Slovenia; UP FAMNIT, Glagoljaska 8, 6000 Koper, Slovenia Received 2 February 2015, accepted 10 October 2015, published online 22 August 2016 Abstract Let P be a polyhedron whose boundary consists of flat polygonal faces on some compact surface S(P) (notnecessarily homeomorphic to the sphere S2). Let voR(P),eoR(P), foR(P) be the numbers of rotational orbits of vertices, edges and faces, respectively, determined by the group G = GR (P) of all the rotations of the Euclidean space E3 preserving P. We define the rotational orbit Euler characteristic of P as the number Eor(P) = voR(P) - eoR(P) + foR(P). Using the Burnside lemma we obtain the lower and the upper bound for EoR(P) in terms of the genus of the surface S(P). We prove that EoR G {2,1,0, -1} for any convex polyhedron P. In the non-convex case EoR may be arbitrarily large or small. Keywords: Polyhedron, rotational orbit, Euler characteristic. Math. Subj. Class.: 52B05, 52B10 1 Introduction CONTEXT: Euler (1752) discovered the famous relation v - e + f = 2 between the numbers of vertices v, edges e and faces f of any convex polyhedron. This Euler polyhedron formula was implicitly stated in the formulas of Descartes (1630) p = 2f + 2v - 4,p = 2e, where p is the number of "plane angles" - corners of faces determined by pairs of adjacent edges ([5], p.469). The number x = v - e + f can be defined for any map (a graph cellularly embedded into a compact surface S) and is called its Euler characteristic. It is related to the genus g of the surface S as follows: x = 2 - 2g ([5], p. 473.) and it may be used for the *This work is supported in part by the Slovenian Research Agency (research program P1-0294 and research project N1-0032). E-mail address: jurij.kovic@siol.net (Jurij Kovic) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 24 Ars Math. Contemp. 13 (2017) 15-21 classification of surfaces by two parameters: one is x and the other is orientability (or non-orientability) of the surface. In [4] we introduced the concept of Euler orbit characteristic Eo = vo — eo + fo, where vo, eo, fo denote the number of orbits of vertices, edges and faces, respectively, determined by the group of all rotations and reflections of the Euclidean space E3, preserving the polyhedron P, and we used it for the classification of the 92 Johnson solids ([4], p.258). In this paper we introduce a similar concept, called rotational orbit Euler characteristic, and we use it for the classification of convex polyhedra. Definition 1.1. The rotational Euler orbit characteristic of the polyhedron P is defined as the number EoR = voR — eoR + foR where voR, eoR, foR are the numbers of rotational orbits of the vertices, edges and faces, respectively, of P (these orbits are determined by the group GR(P) of all the rotations of the Euclidean space E3 preserving P). Proposition 1.2. EoR = 1 for all the Platonic solids and all the n-prisms and n-anti-prisms, while 1 < EoR < 2 for all the Archimedean solids. Proof. In the Table 1 the number of rotational orbits of Platonic and Archimedean solids are given. These values can be easily found for each solid directly or deduced from the symmetry-type graphs of Platonic and Archimedean solids [3]. The 5 Platonic solids have just one rotational orbit of vertices, edges and faces. The 13 Archimedean solids have at most two rotational orbits of vertices and at most three rotational orbits of edges and faces. The n-prisms and the n-antiprisms have just one rotational orbit of vertices and two rotational orbits of edges and faces. □ class solid P vertex pattern VOR eoR for Eor I. tetrahedron (3.3.3) 1 1 1 1 I. octahedron (3.3.3.3) 1 1 1 1 I. cube (4.4.4) 1 1 1 1 I. icosahedron (3.3.3.3.3) 1 1 1 1 I. dodecahedron (5.5.5) 1 1 1 1 II. cuboctahedron (3.4.3.4) 1 1 2 2 II. icosidodecahedron (3.5.3.5) 1 1 2 2 III. truncated tetrahedron (3.6.6) 1 1 2 2 III. truncated cube (3.8.8) 1 1 2 2 III. truncated octahedron (4.6.6) 1 1 2 2 III. truncated dodecahedron (3.10.10) 1 1 2 2 III. truncated icosahedron (5.6.6) 1 1 2 2 IV. rhombicuboctahedron (3.4.4.4) 1 2 3 2 IV. rhombicosidodecahedron (3.4.5.4) 1 2 3 2 V. truncated cuboctahedron (4.6.8) 2 3 3 2 V. truncated icosidodecahedron (4.6.10) 2 3 3 2 VI. snub cube (3.3.3.3.4) 1 3 3 1 VI. snub dodecahedron (3.3.3.3.5) 1 3 3 1 VII. n-prism (4.4.n) 1 2 2 1 VIII. n-antiprism (3.3.n) 1 2 2 1 Table 1: Values of voR, eoR, foR for Platonic and Archimedean solids and for the infinite families of n-prisms and n-antiprisms. MOTIVATION: Similar bounds on EoR exist for the Johnson solids (i.e. convex polyhedra with regular polygonal faces and at least two orbits of vertices [2]). The direct motivation J. Kovic: Classification of convex polyhedra by their rotational orbit Euler characteristic 25 for writing this paper came from the empirical observation that the values of EoR for the 92 Johnson solids are in a small range between -1 and 2. This was discovered during the process of constructing a table of 16 parameters of the Johnson solids presented in [4], while the range for Eo for the same solids turned out to be bigger: 0 < Eo < 5. COMPARISON OF Eo AND EoR: The two characteristics behave very differently on the set of all convex polyhedra: the main result of the paper (Theorem 2.1) states that the relation — 1 < EoR < 2 holds for all convex polyhedra, while for Eo there is no fixed upper bound, we can obtain only the following estimate: Eo = vo — eo + fo < vo + fo < voR + foR = < (vor — eoR + foR) + eoR = eor + eoR < 2 + eoR. Definition 1.3. Let GR(P) = [Ri, R2,..., Rn-i, Rn = Id} be the group of rotational symmetries of the polyhedron P. The poles of the rotation Ri are the points in which the axis of the rotation Ri intersects the surface S(P). Let vp(Ri), ep(Ri), fp(Ri) denote the numbers of poles of Ri in the vertices, edge centers and face centers of P, respectively. The number Ep(Ri) = vp(Ri) — ep(Ri) + fp(Ri) is called the Euler polar characteristics of the rotation Ri. Lemma 1.4. Let ni denote the number of poles of any non-trivial rotation Ri of the polyhedral map P on the surface S of genus g. Then ni < 2(g + 1), m e [0, 2, 4,..., 2(g + 1)}, vp(Ri) + ep(Ri) + fp(Ri) = ni, 0 < ep(Ri) < ni, Ep(Ri) = ni — 2ep(Ri), —2(g + 1) < —ni < Ep(Ri) < ni < 2(g + 1). If the order of the rotation Ri is greater than 2 (i.e. Rn = id and n > 2), then ep(Ri) = 0 and Ep(Rj) = ni. Proof. Any line intersecting P has at most 2g intersecting points with S. If P is a convex polyhedron then any nontrivial rotation Ri has exactly two poles, hence ni = 2. If ni > 2 then each segment of the rotational axis ri not lying in the interior of P contributes two poles (hence ni is an even number!) and at least one new handle. Thus it "increases" the genus of S for 1 (since it is well known that the genus counts the numbers of "handles" of a surface), therefore it must be ni < 2(g +1). The poles can be only in vertices, edge centers or face centers, hence vp(Ri) + ep(Ri) + fp(Ri) = ni and 0 < ep(Ri) < ni. Obviously Ep(Ri) = vp(Ri) — ep(Ri) + fp(Ri) = ni — 2ep(Ri). Therefore the upper bound for Ep(Ri) is ni and the lower bound is —ni. □ Corollary 1.5. If P is a convex polyhedron, then Ep(Ri) = 2 — 2ep (Ri) e [2,0, —2}. If ep =0 then Ep = 2, if ep = 1 then Ep = 1, and if ep = 2 then Ep = —2. If the order of the rotation Ri is greater than 2, then Ep(Ri) =2. Proof. Every convex polyhedron is homeomorphic (by a radial projection from any point of its interior) to a sphere, which has genus g = 0. Now ni = 2 and the formulas follow from the Lemma 1.4. □ The next tool we need (in order to prove the main result, Theorem 2.1) is the Burnside lemma, a standard tool for calculating the number of orbits. 26 Ars Math. Contemp. 13 (2017) 15-21 Lemma 1.6. (Burnside lemma) Let a group G act on some set Q. Let |G| = n denote the number of elements of G and let lFix(g)l denote the number of elements a of the set Q, preserved by the given element g of the group: g(a) = a. Then the number of orbits Qo of the set Q is given by the formula Qo = G E IFix(g)l- |G| geo If the group of rotational symmetries of a convex polyhedron is the cyclical group Cn then the exact value for the Euler orbit characteristic EoR (P) can be obtained by a straightforward application of the Burnside lemma. This is a generalization of the similar result for the spherical polyhedra ([4], p.253). Proposition 1.7. Let P be a convex polyhedron. If GR(P) = Cn where Cn is generated by a rotation R1 (thus Rn = I), then EoR(P) G {0,1,2}. Proof. The identity transformation fixes all vertices, edges and faces while the other n — 1 rotations fix only the poles. Hence we get by the Burnside lemma and using the Euler formula v — e + f = 2 (valid for any convex polyhedron) the following formulas for the numbers of rotational orbits: vor = ~(v + (n — 1)vp), n eoR = —(e + (n — 1)e), n foR = —(f + (n — 1)fp), n Eor =1(2+(n — 1)Ep(Ri)), n and using Ep(R\) = 2 — 2ep(R\) we see: if ep(R\) = 0 then Ep(R\) = 2 and EoR = 2; if ep(Ri) = 1 then n = 2, Ep(Rx) = 0 and EoR = 1; if ep(R1) = 2 then n = 2, Ep(Rl) = —2 and EoR = 0. Thus, if n> 2 then EoR = 2. □ 2 The main result Theorem 2.1. Let P be a polyhedron with faces on the surface S of genus g. Then 1 n— 1 Eor(P) = n(x(P) + E Ep(Ri)), i=i and we get the following bounds on EoR(P): n—1 n—1 1(x(P) — E 2(g +1)) < eor(p) < 1(x(P) + E 2(g +1)). n z—' n z—' i=1 i=1 If P is a convex polyhedron, then — 1 < EoR(P) < 2. J. Kovic: Classification of convex polyhedra by their rotational orbit Euler characteristic 27 Proof. Let n be the number of elements in the group GR(P). The identity transformation fixes each vertex, edge or face. Every rotation Ri fixes vp(Ri) vertices, ep(Ri) edges and fp(Ri) faces. Therefore, by the Burnside lemma: VOR = 1 (v + Vp(R 1) +----Vp(Rn-1)), eon = —(e + ep(R 1) +----ep(Rn-1)), n fOR =—(f + fp(R1) + ••• fp (Rn — 1)), n Eon = —(X + Ep(R1) + ■ ■ ■ Ep(Rn—1)). Using -2(g +1) < Ep(Ri) < 2(g + 1) (proved in Lemma 1.4) we get n—1 n—1 — (x - V 2(g + 1)) < Eon < — (x + V 2(g + 1)). nn i=1 i=1 If P is a convex polyhedron, then g = 0, x = 2, hence Eor < 1 (2 + (n - 1)2) = 2, n 114 Eor > 1(2 + (n - 1)(-2)) = 1(4 + n(-2)) > -2 + - > -1, nnn because n > 0 and EoR must be an integer. □ Thus there are 4 classes C2, C1, Co, C—1 of convex polyhedra, whose Eor are 2,1, 0, -1, respectively. Is the lower bound EoR = -1 actually obtained, and (if it is so) for which convex polyhedra? And is there any simple description of these four classes? Proposition 2.2. Let a, b, c be the numbers of rotations R in the group GR(P) of a convex polyhedron for which Ep(Ri) equals 2, 0 and -2, respectively, and let n be the number of elements in GR(P). Then 12 Eor(P) = 1(2 + a ■ 2 + c(-2)) = -(1 + a - c). nn Thus the number 1 + a - c is an integer multiple of n. The numbers a and c can be (for each of the 4 possible values of EoR) expressed by b, n and EoR. Proof. This formula follows immediately from the Burnside lemma. Also, it is clear that a + b + c +1 = n, hence a + c = n - b - 1. 28 Ars Math. Contemp. 13 (2017) 15-21 The equation n(1 + a - c) = EoR implies 2(1 + a - c) = EoR ■ n and EOr ■ n c — a = 1 — 2 Then (a + c) + (c - a) = 2c = n - b - 1 + 1 - ^^p = (2-E°°R)n - b and _ (2 - EoR)n - 2b c = 4 • Similarly, 2a = n - b - 1 - 1 + ^^, hence n ■ (2 + Eor) - (2b + 4) a =-4-• For example, if EoR = -1 and b = 0 then a = n - 1 and c = ^n -In that case n must be divisible by 4. □ Example 2.3. To find such a solid with 4 symmetries we have to look for one having three rotations with poles in edge centers! The lower bound EoR = -1 is really obtained for the Johnson solid J84 (Snub Disphenoid, see Figure 1), where voR = 2, eoR = 6, foR = 3, hence EoR = 2 - 6 + 3 = -1. Here the number of symmetries is 4 (the identity transformation and 3 rotations of order two with axes going through edge centers), b = 0, a = 0 and c = 3. Thus this lower bound -1 is sharp. Figure 1: The Johnson solid J84, also known as the Snub Disphenoid. Remark 2.4. A rotational axis of a non-convex polyhedron may have more than 2 "poles". As a consequence, there is no upper or lower bound for EoR in the non-convex case. 3 Classification of convex polyhedra As an immediate consequence of the formulas in Proposition 3 we get: J. Kovic: Classification of convex polyhedra by their rotational orbit Euler characteristic 29 Corollary 3.1. The four classes C2,C1,C0,C-1 of convex polyhedra (whose EoR are 2,1,0, — 1, respectively) can be characterized as follows: Ci: a — c = n — 1 C1: a — c = —1 + n Co: a — c = —1 C-1: a — c = —1 — n, all poles in edge centers, a = 0. Corollary 3.2. If a > n/2 then EoR e {1,2}. Proof. The relation a > n/2 is a sufficient condition for a — c > 0 (since there is also the identity transformation in the group GR (P)) that holds only for polyhedra from C2 and Ci. □ Corollary 3.3. Let q be the number of all rotations R e GR(P) with the property that R has the same rotational axis as some k-fold rotation for any k > 2. If q > n/2, where n is the order of the group GR, then EoR(P) e {1, 2}. Proof. No rotation with such an axis can have any of its two poles in an edge center, hence a > q > n/2, therefore EoR(P) e {1, 2}. □ Now we can classify convex polyhedra with respect to their rotational symmetry groups and their rotational orbit Euler characteristic. The only possible rotational groups of the Euclidean space E3 are the rotational groups of 1) the n-gonal pyramid, 2) the n-gonal dipyramid or prism, 3) the regular tetrahedron, 4) the cube or the regular octahedron, 5) the regular dodecahedron or the regular icosahedron ([1], p.34). Theorem 3.4. Convex polyhedra with at least one rotational symmetry can be classified by their GR and by their EoR into 13 classes (in Table 2 the impossible cases are marked with $): C2 Ci Co C_i Cn cyclical group 0 Dn dihedral group T tetrahedron group 0 0 O octahedron group 0 0 D dodecahedron group 0 0 Table 2: Classification of convex polyhedra by GR and EoR. Proof. If Gr = Cn then EoR = -1 (by Proposition 2). If GR G {T, O, D} then q > n/2 (Table 3) hence P cannot be in C0 or C_i (by Corollary 3.3). □ 30 Ars Math. Contemp. 13 (2017) 15-21 2 3 4 5 n q T tetrahedron 3 4 3 + 4.2 + 1 = 12 9 O cube 6 4 3 6 + 4.2 + 3.3 + 1 = 24 17 D dodecahedron 15 10 6 15 + 10.2+12.4 + 1= 60 44 Table 3: Numbers of the 2-,3-,4-,5-fold axes in the solids P with the rotational groups T, O or D, orders n of the groups T, O, D and the numbers q of all rotations R G GR(P) with the property that R has the same rotational axis as some k-fold rotation for any k > 2. References [1] H. S. M. Coxeter, W.O.J. Moser, Generators and Relations for Discrete Groups, Springer Verlag, Berlin, Gottingen, Heidelberg, 1957. [2] N. W. Johnson, Convex polyhedra with regular faces, Canad. J. Math. 18 (1966), 169-200. [3] J. Kovic, Symmetry-type graphs of Platonic and Archimedean solids, Math. Commun. 16 (No 2) (2011), 491-507. [4] J. Kovic, Characterization of convex polyhedra with regular polygonal faces by minimal number of parameters, J. Combin. Math. Combin. Comput. 89 (2014), 249-263. [5] J. Stillwell, Mathematics and Its History, Springer, New York, 2010. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 31-47 Lifting symmetric pictures to polyhedral scenes Viktoria E. Kaszanitzky * Department of Operations Research, Eotvos University, Pâzmâny Péter sétâny 1/C, 1117 Budapest, Hungary and Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom Bernd Schulze f Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF, United Kingdom Received 23 June 2015, accepted 21 May 2016, published online 6 October 2016 Scene analysis is concerned with the reconstruction of d-dimensional objects, such as polyhedral surfaces, from (d - 1)-dimensional pictures (i.e., projections of the objects onto a hyperplane). In this paper we study the impact of symmetry on the lifting properties of pictures. We first use methods from group representation theory to show that the lifting matrix of a symmetric picture can be transformed into a block-diagonalized form. Using this result we then derive new symmetry-extended counting conditions for a picture with a non-trivial symmetry group in an arbitrary dimension to be minimally flat (i.e., 'non-liftable'). These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the various symmetry operations of the picture. We then also transfer lifting results for symmetric pictures from Euclidean (d - 1)-space to Euclidean d-space via the technique of coning. Finally, we offer some conjectures regarding sufficient conditions for a picture realized generically for a symmetry group to be minimally flat. Keywords: Incidence structure, picture, polyhedral scene, lifting, symmetry, coning. Math. Subj. Class.: 68T45, 20C99, 52C25 * Research was supported by the EPSRC First Grant EP/M013642/1 and by the Hungarian Scientific Research Fund (OTKA, grant number K109240). tResearch was supported by the EPSRC First Grant EP/M013642/1. E-mail addresses: viktoria@cs.elte.hu (Viktoria E. Kaszanitzky), b.schulze@lancaster.ac.uk (Bernd Abstract Schulze) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 32 Ars Math. Contemp. 13 (2017) 15-21 1 Introduction An important and well studied problem in Artificial Intelligence, Computer Vision and Robotics is to find efficient methods for determining whether a straight line drawing in the Euclidean plane (also known as a '2-picture') corresponds to the projection of a 3-dimensional polyhedral surface (also known as a polyhedral '3-scene'). Applications of these results include image understanding, monocular vision and automatic reconstructions of 3-dimensional polyhedral objects or environments. In the Computer Vision community, this problem was first studied by Mackworth and Huffman [12, 7]. Using 'labeling schemes' and 'reciprocal diagrams', they obtained necessary conditions for the realizability of 2-pictures as polyhedral 3-scenes. However, the geometric method of the 'reciprocal diagram' has already been used by J. C. Maxwell and L. Cremona in the 19th century as a graphical tool to analyze the statics of trusses and mechanical structures [5, 13]. This work provides a beautiful connection between the field of polyhedral scene analysis and the field of static (or equivalently, infinitesimal) rigidity of frameworks [27, 29]. For further connections between these fields and other areas of discrete geometry, such as parallel redrawings of configurations (which is the dual interpretation of liftings of pictures), Minkowski decomposability of polytopes, and projective point-line configurations, see [21, 33, 34] for example. While reciprocal diagrams provide a powerful tool to check for inconsistencies in pictures, they do not provide sufficient conditions for the realizability of pictures as polyhedral scenes. In [22,23,24] Sugihara used linear programming methods to establish both a necessary and sufficient condition for a general picture to represent a polyhedron (see also [25]). Various other necessary and sufficient conditions were subsequently obtained by Crapo, Whiteley, et al. using a variety of different methods ranging from projective geometry and Grassmann-Cayley algebra to invariant theory (see e.g. [3, 4, 14, 26, 30, 31, 32]). A fundamental tool for analyzing a given picture is the lifting matrix, whose rank, row dependencies and column dependencies completely determine the liftability properties of the picture (see e.g. [3, 31, 33, 34]). In particular, this matrix yields some simple necessary counting conditions for a (d — 1)-dimensional picture to be 'flat' (i.e., non-liftable to a d-dimensional polyhedral scene) in terms of the number of vertices, faces and incidences of the underlying combinatorial incidence structure. Following a conjecture of Sugihara [23], Whiteley showed in [31] that these counts are also sufficient for 'generic' pictures (with the same underlying incidence structure) to be flat. In this paper, we study the impact of symmetry on the lifting properties of (d — 1)-dimensional pictures. This has important practical applications since symmetry is ubiquitous in both man-made and natural structures. Moreover, there has recently been a surge of interest in studying the impact of symmetry on the static or infinitesimal rigidity properties of structures (see e.g. [2, 6, 10, 15, 18, 19]), and hence it is natural to apply similar group-theoretic methods to the lifting analysis of symmetric pictures. In Section 4 we first show that the lifting matrix of a symmetric (d — 1)-picture can be transformed into a block-diagonalized form using methods from group representation theory. This is a fundamental result, since the block-decomposition of the lifting matrix can be used to break up the lifting analysis of a symmetric picture into a number of independent subproblems, one for each block of the lifting matrix. In fact, the analogous result for the rigidity matrix of a symmetric framework (see [10,15]) is basic to most of the recent results regarding the rigidity analysis of symmetric structures (see e.g. [2, 6, 18]). Similarly to [15], the block-decomposition of the lifting matrix is obtained by showing that V. E. Kaszanitzky and B. Schulze: Lifting symmetric pictures to polyhedral scenes 33 (a) (b) (c) Figure 1: A (minimally) flat 2-picture (where all four interior regions are faces) (a) which becomes sharp if realized with reflectional symmetry (b). A non-trivial (and sharp) lifting of the picture in (b) is shown in (c). it intertwines two particular matrix representations of the given symmetry group. For the lifting matrix, one of these representations is associated with the incidences of the picture and the other one is associated with the vertices and faces of the picture (see Theorem 4.1). In Section 5 we then use these results, together with some methods from character theory, to derive new necessary counting conditions for a symmetric picture to be 'minimally flat' (i.e., flat with the property that the removal of any incidence leads to a picture which does have a non-trivial lifting). Such pictures may be thought of as the basic building blocks for symmetric flat pictures, as we may (symmetrically) add further incidence constraints to a minimally flat picture to obtain classes of (over-constrained) flat pictures. We then follow the approach in [2] to derive a complete list of the necessary conditions for 2-dimensional pictures to be minimally flat, as these are the most important structures for practical applications. Similar counts for higher-dimensional pictures can easily be obtained for any symmetry group using Corollary 5.5. A well established tool in rigidity theory for transferring results from an Euclidean space to the next higher dimension (as well as to other types of metrics) is the technique of 'coning' (see e.g. [20, 28]). In the end of Section 5 we show that coning can also be used to transfer lifting results for pictures from (d — 1)-space to d-space. Finally, in Section 6 we offer some conjectures regarding combinatorial characterizations of minimally flat pictures which are as generic as possible subject to the given symmetry constraints. Moreover, we briefly discuss the question of whether a picture which is generic modulo symmetry has a 'sharp' lifting, i.e. a lifting where any two faces sharing a vertex lie in different hyperplanes. 2 Pictures, liftings, and scenes A (polyhedral) incidence structure S is an abstract set of vertices V, an abstract set of faces F, and a set of incidences I C V x F. A (d — 1)-picture is an incidence structure S together with a corresponding location map r : V ^ Rd-1, ri = (xj, Vi,..., wi)T, and is denoted by S(r). A d-scene S(p, P) is an incidence structure S = (V, F; I) together with a pair of location maps, p : V ^ Rd, pi = (xi,...,wi, zi)T, and P : F ^ Rd, Pj = (Aj,..., Cj, Dj )T, such that for each (i, j) e I we have Ajxi +... + Cj wi + zi + Dj = 0. (We assume 34 Ars Math. Contemp. 13 (2017) 15-21 that no hyperplane is vertical, i.e., is parallel to the vector (0,..., 0,1)T.) A lifting of a (d - 1)-picture S(r) is a d-scene S(p, P), with the vertical projection n(p) = r. That is, if p = (xj,..., Wj,Zj)T, then r = (xj,... ,Wj)T = n(p4). A lifting S(p, P) is trivial if all the faces lie in the same plane. Further, S(p, P) is folded (or non-trivial) if some pair of faces have different planes, and is sharp if each pair of faces sharing a vertex have distinct planes. A picture is called sharp if it has a sharp lifting. Moreover, a picture which has no non-trivial lifting is called flat (or trivial). A picture with a non-trivial lifting is called foldable. The lifting matrix forapicture S (r) is the |I1 x (|V | + d|F |) coefficient matrix M (S,r) of the system of equations for liftings of a picture S(r): For each (i, j) G I, we have the equation Aj xj + Bj yj + ... + Cj wj + zj + Dj = 0, where the variables are ordered as [..., zj,...;..., Aj, Bj,..., Dj,...]. That is the row corresponding to (i, j) G I is: (i,j) 0... 0 1 0... 0 0... 0 rj 1 0... 0 |V | d|F | Theorem 2.1 (Picture Theorem). [31, 33] A generic picture of an incidence structure S = (V, F; I) with at least two faces has a sharp lifting, unique up to lifting equivalence, if and only if |I1 = |V| + d|F| - (d + 1) and |I '| < |V'| + d|F'| - (d + 1) for all subsets I' of incidences with at least two faces. A generic picture of S has independent rows in the lifting matrix if and only if for all non-empty subsets I' of incidences, we have |I '| < |V '| + d|F '| — d. Note that it follows from the Picture Theorem that a generic picture of an incidence structure S = (V, F; I) is minimally flat, i.e. flat with independent rows in the lifting matrix, if and only if |I| = |V| + d|F| - d and |I'| < | V'| + d|F'| - d for all non-empty subsets I' of incidences. 3 Symmetric incidence structures and pictures An automorphism of an incidence structure S = (V, F; I) is a pair a = (n,a), where n is a permutation of V and a is a permutation of F such that (v, f) G I if and only if (n(v), a(f)) G I for all v G V and f G F. For simplicity, we will write a(v) for n(v) and a(f) for a(f). The automorphisms of S form a group under composition, denoted Aut(S). An action of a group r on S is a group homomorphism 0 : r ^ Aut(S). The incidence structure S is called r-symmetric (with respect to 0) if there is such an action. For simplicity, if 0 is clear from the context, we will sometimes denote the automorphism 0(y) simply by 7. Let r be an abstract group, and let S be a r-symmetric incidence structure (with respect to 0). Further, suppose there exists a group representation t : r ^ O(Rd-1). Then we say that a picture S(r) is r-symmetric (with respect to 0 and t) if t(Y)(rj) = re(7)(j) for all i G V and all 7 G r. (3.1) In this case we also say that t(r) = {t(7) 17 g r} is a symmetry group of S(r), and each element of t(r) is called a symmetry operation of S(r). Throughout this paper, we will use the Schoenflies notation for symmetry operations and symmetry groups, as this is one of the standard notations in the literature on symmetric structures [2, 6, 10, 15]. V. E. Kaszanitzky and B. Schulze: Lifting symmetric pictures to polyhedral scenes 35 4 Block-decomposing the lifting matrix of a symmetric picture In this section we will show that by changing the canonical bases for R|f| and R|V|+d|F 1 to appropriate symmetry-adapted bases, the lifting matrix of a symmetric picture can be transformed into a block-decomposed form. Let r be an abstract group, and let S = (V, F; I) be a r-symmetric incidence structure (with respect to 0 : r ^ Aut(S)). Further, let S(r) be a r-symmetric (d - 1)-picture with respect to the action 0 and the homomorphism t : r ^ O(Rd-1). We fix an ordering of the vertices in V, the faces in F and the incidences in I. We let Pv : r ^ GL(R|V|) be the linear representation of r defined by Py (y) = [5e(Y)(j)]i,j, where S denotes the Kronecker delta symbol. That is, PV (7) is the permutation matrix of the permutation of V induced by 0(7). Similarly, we let PF : r ^ GL(R|F|) be the linear representation of r defined by PF(7) = [Se(Y)(j)]i,j. That is, PF(7) is the permutation matrix of the permutation of F induced by 0(7). Moreover, note that for each Y € r, the automorphism 0(7) of S clearly also induces a permutation of the incidences I of S. So, analogously to PV and PF, we let Pf : r ^ GL(Ri11) be the linear representation of r which consists of the permutation matrices of the permutations of I induced by 0. We call P/ the internal representation of r (with respect to 0 and t). The external representation of r (with respect to 0 and t) is the linear representation Py 0 (f < PF) :r ^ GL(R|V 1 0 Rd|F|), /t (y) 0 where f : r ^ O(Rd) is the augmented representation of t, definedby f(Y) = ( 0 1 Recall that given two linear representations, p1 : r ^ GL(X) and p2 : r ^ GL(Y) with representation spaces X and Y, a linear map T : X ^ Y is said to be a r-linear map of p1 and p2 if T o p1 (y) = p2 (y) 0 T for all y € r. The vector space of all r-linear maps of p1 and p2 is denoted by Homr(p1, p2). Theorem 4.1. Let S = (V, F; I) be a r-symmetric incidence structure (with respect to 0), let t : r ^ O(Rd-1) be a homomorphism, and let S(r) be a r-symmetric (d — 1)-picture (with respect to 0 and t). Then M(S, r) € Homr(PV 0 (f < PF), P/). Proof: Suppose we have M(S, r)c = b. Then we need to show that for all y € r, we have M(S, r)(Py 0 (f < Pf))(y)c = P/(y)b. Fix y € r, and let 0(y) = (n, a). The automorphism of I induced by 0(y) we denote by a. Let (i, j) € I. Further, let n(i) = k and a(j) = l. Thus, a((i, j)) = (k, l). Note that Pf (y)& is an |1 |x 1 column vector which is indexed by the incidences in I. By the definition of Pf (y), for the (k, l)-th component, (Pf (Y)b)(k,j), of Pf (Y)b we have (Pf (Y)b)(k,;) = (b)(i,j). So we need to show that (M(S,r)(Py 0 (f < Pf))(Y)c)(fc,i) = (b)(i,j). Note that (M(S, r)c)(M) = (b)(M) = zfc + A1 xfc + + ... + + D1. By the definition of Py 0 (f < Pf)(y), (M(s, r)(Py 0 (f < Pf))(y)c)(/m) is equal to zi + (xfc,yfc,...,wfc, 1) ( T0Y) 1 ( Aj \ V D j ) 36 Ars Math. Contemp. 13 (2017) 15-21 By symmetry (recall that n(i) = k, and hence t(7)(r) = rk), this is equal to zi + (*,*,...,«*, 1)( TMT( TJ í Aj \ Bj \D y Since t(7) (and hence also t(y)) is an orthogonal matrix, this is equal to zi + XiAj + ... + WiCj + Dj = (b)(i,j). This completes the proof. □ Since M(S, r) G Homr(PV © (f ( PF),P/), there are non-singular matrices A and B such that BTM(S, r)A is block-diagonalized, by Schur's lemma (see [8] e.g.). If p0,..., pn are the irreducible representations of r, then for an appropriate choice of symmetry-adapted coordinate systems, the lifting matrix takes on the following block form BtM(S, r)A := M(S,r) ( Mo(S,r) 0 (4.1) V 0 Mn(S,r) where the submatrix block Mj(5, r) corresponds to the irreducible representation pi of r. More precisely, the symmetry-adapted coordinate systems can be obtained as follows. Recall that every linear representation of r can be written uniquely, up to equivalency of the direct summands, as a direct sum of the irreducible linear representations of r. So we have Pv © (f PF) = Aopo © ... © A„p„, where Ao,..., A„ G N U {0}. (4.2) For each t = 0,..., n, there exist At subspaces (V(pt))p..., (V(pt))A of the R-vector space R|V|+d|F 1 which correspond to the At direct summands in (4.2), so that r|v |+d|F 1 = v (po) © ... © V (4.3) where V(pt) = (V(pt))1 © ... © (V(pt))At. (4.4) Similarly, for the internal representation PI of r, we have the direct sum decomposition Pi = MoPo © ... © MnPn, where Mo,..., Mn G N U {0}. (4.5) For each t = 0,..., n, there exist Mt subspaces (W(pt)) 1,..., (W(pt)) of the R-vector space R111 which correspond to the Mt direct summands in (4.5), so that R111 = W(po) © ... © W(4.6) where ( ) ( ) W(pt) = (W(It))1 © ... © (W(pt))Mt. (4.7) If we choose bases (A(pt)) 1,..., (A(pt)) for the subspaces in (4.4) and we also choose bases (B(pt))1,..., (B(pt))^ for the subspaces in (4.7), then U=o 1 (A(pt))i and IXoll1 (B(pt))i are symmetry-adapted bases with respect to which the lifting matrix is block-decomposed as shown in (4.1). V. E. Kaszanitzky and B. Schulze: Lifting symmetric pictures to polyhedral scenes 37 Definition 4.2. A vector c G R|vl+dlF 1 is symmetric with respect to the irreducible linear representation pt of r if c G V(pt). Similarly, we say that a vector b G R11 is symmetric with respect to pt if b G W(pt). Note that the kernel of the block matrix Mt(S,r) is isomorphic to the space of all liftings of the r-symmetric picture S(r) which are symmetric with respect to pt. 5 Symmetry-extended counting rules for the foldability of pictures Recall from the Picture Theorem (Theorem 2.1) that if S(r) is minimally flat, then it satisfies |11 = |V| + d|F| - d. Clearly, if |11 < |V| + d|F| - d, then S(r) has a non-trivial lifting, and if |11 > |V| + d|F| — d, then the lifting matrix M(S,r) has a non-trivial row dependence. In Section 5.1, we will use the results of the previous section to derive a symmetry-extended version of this formula, which will provide further necessary counting conditions for a symmetric picture in an arbitrary dimension to be minimally flat. As we will see, these conditions can be stated in a very simple way in terms of the numbers of structural components of the picture that are fixed by the various symmetry operations. In Section 5.2 we will then derive a complete list of the necessary counting conditions for symmetric pictures in the plane to be minimally flat. Finally, in Section 5.3 we consider the transfer of lifting results for pictures from (d — 1)-space to d-space via coning. 5.1 Symmetry-extended counting rules Recall that if p : r ^ GL(X) is a linear representation of a group r with representation space X then a subspace U of X is said to be p-invariant if p(j)(U) C U for all y G r. Proposition 5.1. Let S(r) be a picture which is r-symmetric with respect to 9 : r ^ Aut(S) and t : r ^ Rd-1. Then the space T (S, r) of trivial liftings of S(r) is a Pv © (T < PF)-invariant subspace of R|v 1 © Rd|F|. Proof: Let t be any element of T(S,r). Then t is an element of the kernel of the lifting matrix M (S, r) of the form (..., zi,...,..., Aj ,Bj ,...,Dj,.. .)T, where Aj = Ak, Bj = Bk, ..., Dj = Dk for all 1 < j,k < |F|. It follows from Theorem 4.1 that (Pv © (t < PF))(t) is also an element of the kernel of M(S,r), and it follows immediately from the definition of Pv © (T < PF) that (Pv © (t < PF))(t) is of the form (...,zi,...,..., Aj, Bj,..., Dj,.. .)T, where Aj = A'k, Bj = B'k, ..., Dj = D'k for all 1 < j,k < |F |. This gives the result. □ We denote by (Pv © (T < PF))(T) the subrepresentation of Pv © (T < PF) with representation space T(S,r). Recall that the character of a representation p : r ^ GL(X) is the row vector x(p) whose i-th component is the trace of p(^i), for some fixed ordering Y1,..., 7|r| of the elements of r. Theorem 5.2 (Symmetry-extended counting rule). Let S(r) be a (d — 1)-picture which is r-symmetric with respect to 9 and t. If S(r) is minimally flat, then we have x(Pi) = x(Pv © (T < Pf)) — x((Pv © (T < Pf))(T)). (5.1) Proof: By Maschke's Theorem (see [8] e.g.), T(S, r) has a (Pv © (t < PF))-invariant complement Q in R|v\+d\F|. We may therefore form the subrepresentation (Pv © (T < 38 Ars Math. Contemp. 13 (2017) 15-21 PF))(Q) of PV © (f < PF) with representation space Q. Since (S, r) is minimally flat, the restriction of the linear map represented by the lifting matrix M (S, r) to Q is an isomorphism onto R11|. Moreover, since M(S, r) is T-linear with respect to the representations PV ©(f ^t, then, by (4.1), there exists a non-trivial lifting of S(r) (i.e., a non-trivial element in the kernel of M(S, r)) which is symmetric with respect to pt, and if Kt < then there exists a nontrivial row dependence of M(S, r) (i.e., a non-zero element in the co-kernel of M(S, r)) which is symmetric with respect to pt. Before we illustrate this symmetry-adapted counting rule by means of an example, we show how the characters in (5.1) can be computed in a very simple way. We need the following definitions. Let S be an incidence structure and let 0 : r ^ Aut(S) be a group action on S. A vertex v of S is said to be fixed by 7 e r (with respect to 0) if 7« = v. Similarly, a face f = {vi,..., vm} of S is said to be fixed by 7 e r (with respect to 0) if 7/ = f, i.e., if 7 ({v1,..., vm}) = {v1,..., vm}. Finally, an incidence (i, j) of S is said to be fixed by 7 e r (with respect to 0) if Y((i, j)) = (i, j) The sets of vertices, faces, and incidences of a r-symmetric incidence structure S which are fixed by 7 e r are denoted by VY, FY, and IY, respectively. Remark 5.3. Note that if a (d - 1)-picture S(r) is r-symmetric (with respect to 0 and t) and a vertex i is fixed by some 7 e r, then r4 must occupy a special geometric position in Rd-i. For example, if t (7) is a reflection in the plane, then r must lie in the corresponding mirror line. Similarly, if t(7) is a rotation in the plane, then r must lie at the center of rotation (i.e., the origin). Similar geometric restrictions are imposed on any faces (or incidences) of S(r) that are fixed by an element 7 e r. V. E. Kaszanitzky and B. Schulze: Lifting symmetric pictures to polyhedral scenes 39 Proposition 5.4. Let r = {71,..., Y|r|} be an abstract group and let S(r) be a (d — 1)-picture which is T-symmetric with respect to 9 and t. Then we have (i) x(Pi) = (|/711,...,|i7|r|I); (ii) x(Pv 0 (r << Pf )) = (|V7i | + tr(f(7i)) |F7i |,..., |V7 | r 11 + tr(r(7|r|)) |F7| r 11); (iii) x((Pv 0 (r << Pf))(t)) = x(r). Proof: (i) Note that tr(Pi(7)) = |/71 for each 7 G r. (ii) Similarly, we have tr(PV(y)) = V1, tr(PF(7)) = |F71, and tr((r ® Pf)(y)) = tr(r(7))tr(PF(7)) for each 7 G r. (iii) A basis for the space T(S, r) of trivial liftings of S(r) is given by the d vectors Ti = (xi, ... ,X|V|, —ei, .. ., —ei)T, T2 = (yi, ... ,y|y|, — e2,. .., —e2)T, .. ., Td_i = (wi,...,w|v |, —ed-i,..., — ed-i)T and Td = (1,..., 1, — ed,..., —ed)T, where e4 denotes the ¿-th canonical basis vector of Rd. An elementary calculation shows that for every 7 G r, we have (Pv 0 (r Pf))(T)(7)Tj = (r(7))ijTi + • • • + (r(7))djTd. This gives the result. □ By Proposition 5.4, the symmetry-extended counting rule in (5.1) can be simplified to x(pi)= x(Pv 0 (r < pf)) — x(r), (5.2) and each of these characters can be computed very easily. (The calculations of the characters x(PI), x(PV 0 (r < PF)), and x(r) for pictures in dimension 2 are shown in Table 2.) Moreover, note that this vector equation gives one equation for each element of the group r. This leads to the following very useful corollary of Theorem 5.2, which allows us to detect non-trivial liftings in symmetric pictures by simply counting the number of structural components of the picture that are 'fixed' by a given symmetry operation of the picture. Corollary 5.5. Let (S, r) be a (d — 1)-picture which is T-symmetric with respect to 9 and t . If (S, r) is minimally flat, then for each 7 G r, |Iy | = |V71 + ir(r(7))(|F71 — 1). (5.3) Proof: This follows immediately from Theorem 5.2 and Proposition 5.4. □ The following example illustrates how to apply the symmetry-extended counting rule to a picture in dimension 2. Example 5.6. Consider the 2-picture with the reflectional symmetry group Cs = {¿d, s} in Figure 2. For this picture, we have |V| = 6, |F| = 4, |I| = 15, and hence |I| = |V| + 3|F| — 3 = 15. So, for generic positions of the vertices, we obtain flat pictures. However, using our symmetry-extended counting rule we can easily show that the mirror symmetry shown in Figure 2 induces a non-trivial lifting. The group Cs has two non-equivalent irreducible representations po and pi, each of which is of dimension 1 (see Table 1). Since tr(r(id)) = 3, tr(r(s)) = 1, |V | +3|F | = 18, and |VS| + |FS| = 2 + 2 = 4, we have x(Pv 0 (r < Pf)) — X((PV 0 (r < PF))(T)) = (18, 4) — (3,1) = (15, 3) = 9po + 6pi. 40 Ars Math. Contemp. 13 (2017) 15-21 Cs id s po 1 1 pi 1 -1 Table 1: The characters of the irreducible representations of the group Cs. Further, since |11 = 15 and |/s | = 1, we have x(Pi) = (15,1) = 8po + 7pi. Thus, we may conclude that the picture in Figure 2 has a non-trivial lifting which is symmetric with respect to p0 (i.e., the lifting preserves the mirror symmetry) and a row dependence which is symmetric with respect to p1. Note that for this particular example, we could also have used the results in [2] to see that the corresponding bar-joint framework has a self-stress, and then deduce the existence of a (sharp) lifting via the technique of the reciprocal diagram [13, 12]. Figure 2: A 2-picture with mirror symmetry which has a symmetry-induced non-trivial lifting (see also Figure 1(c)). 5.2 When is a symmetric 2-picture minimally flat? The possible non-trivial symmetry operations of a picture in dimension 2 are reflections in lines through the origin (denoted by s), and rotations about the origin by an angle of , where n > 2 (denoted by Cn). Therefore, the possible symmetry groups in the plane are the identity group C1, the rotational groups Cn, n > 2 (generated by a single rotation Cn), the reflection group Cs (generated by a single reflection s), and the dihedral groups Cnv, n > 2. In Table 2, we show the calculations of characters for the counting condition in (5.1) (or equivalently, (5.2)) for 2-pictures. In this table, |Vn|, |Fn|, and |/n| denote the numbers of vertices, faces, and incidences that are fixed by an n-fold rotation Cn, n > 2, respectively. Similarly, | Vs |, |Fs |, and |/s | denote the numbers of vertices, faces, and incidences that are fixed by a reflection s, respectively (recall also the notation introduced in Subsection 5.1). From equation (5.3) and Table 2, we obtain the following necessary conditions for a r-symmetric 2-picture (with respect to 0 and t) to be minimally flat. If x(PI) = x(Pv ® (f PF)) - x(f),then V. E. Kaszanitzky and B. Schulze: Lifting symmetric pictures to polyhedral scenes 41 id Cn>2 C2 s x(Pi ) |11 |1n| |12| |1s| x(Pv ® (T ® Pf)) |V | +3|F | |V„| + (1 + 2 COS 2f )|F„| |V2| — |F2| |Vs| + |Fs| X(r) 3 1 + 2 cos ^^ n —1 1 Table 2: Calculations of characters for the symmetry-extended counting rule for minimally flat pictures in dimension 2. id: |11 = |V| + 3|F|- 3 (5.4) C2: |121 = |V2|-|F2| + 1 (5.5) s: |1s| = |VS| + |FS| — 1 (5.6) 2n C„>2: |/„| = |Vn| + (|Fn| — 1)(1 + 2cos — ) (5.7) n where a given equation applies when the corresponding symmetry operation is present in t(r). Some observations on minimally flat 2-pictures, arising from this set of equations are: 1. Every symmetry group contains the identity id, and hence we must always have the standard count |11 = | V| + 3|F| — 3. 2. It follows from (5.5) that the presence of a half-turn C2 imposes limitations on the placements of vertices and faces. Note that if |V2| =0 or |F2| = 0, then |/2| = 0. Thus, we must have |F21 > 0. Also, if |V2| = 0, then we must have |F2| = 1. If |V2| = 1 then by |121 < |F21 either |121 = |F2| = 1 or |121 = 0, |F2| = 2 holds. 3. Similarly, by (5.6), presence of a mirror line implies that if | Vs | = 0, then |Fs | = 1, and if |FS| =0, then |VS| = 1. 4. By (5.7), presence of a rotation of higher order Cn>2 gives rise to the following conditions. If n = 3, then the equation in (5.7) becomes |131 = |V3|. If n = 4, then the equation in (5.7) becomes |/4| = |V4| + |F4| — 1. Therefore, if |V4| = 0, then |F4| = 1 and, if |V4| = 1 then |/4| = |F4| < 1 by |/4| < |/2|. If n = 6, then the equation in (5.7) becomes 11 = |V6| + 2|F6| — 2. Therefore, | F61 > 0 (for otherwise, | V61 = 2 and the location map of the picture would be non-injective). Further, if |V6| = 0, then |Fe| = 1 and, if |V6| = 1 then = F| = 1 holds by |/6| < |121. Finally, if Cn is present, where n ^ {2, 3,4,6}, we must have |Fn| = 1 and |/„| = |Vn|. Similarly, we may obtain lists of necessary conditions for symmetric pictures in 3- or higher-dimensional space to be minimally flat (using Corollary 5.5). 42 Ars Math. Contemp. 13 (2017) 15-21 C2 C3 C4 Cs Figure 3: Some symmetric minimally flat 2-pictures (where all interior regions are faces). C2 Cs C 3v Figure 4: Some symmetric 2-pictures with a (sharp) symmetry-induced lifting (where all interior regions are faces). 5.3 Coning (d — 1)-pictures In the following we show that the technique of 'coning' can be used to construct (minimally) flat r-symmetric d-pictures from (minimally) flat r-symmetric (d — 1)-pictures. Let S = (V, F; I) be an incidence structure and let S(r) be a (d — 1)-picture for d > 2. The coned incidence structure S = (V, F; I) is obtained by adding a new vertex v to V, replacing each face f e F by f U {v}, and adding the incidences (v, f), f e F. A realization of S as a d-picture S(r ) is called a coned picture of S(r). An example of a coned 2-picture is shown in Figure 5. Let r be a group, and let S(r) be a r-symmetric (d—1)-picture (with respect to 0 and t) with n vertices. Then S(r) is said to be r-generic if the set of coordinates of the image of r are algebraically independent over Qr, where Qr denotes the field generated by Q and the entries of the matrices in t(r). In other words, S(r) is r-generic if there does not exist a polynomial h(x 1,..., £(d_i)n) with coefficients in Qr such that h((ri)i,..., (r„)d_i) = 0. (Note that if r is the trivial group, then Qr = Q. In this case, a r-generic picture is simply called generic.) Clearly, the set of all r-generic realizations of S is a dense (but not open) subset of all r-symmetric realizations of S. Moreover, all r-generic realizations of S share the same lifting properties. We say that S is r-generically (minimally) flat in Rd-1 if all r-generic realizations of S in Rd-1 are (minimally) flat. For a r-symmetric (d — 1)-picture S(r) (with respect to 0 : r ^ Aut(S) and t : r ^ O(Rd-1)), we let I : r ^ O(Rd) be the augmented representation, i.e., I(7) = V E. Kaszanitzky and B. Schulze: Lifting symmetric pictures to polyhedral scenes 43 "(7) . Moreover, for the coned incidence structure S = (V, F; /) (with cone vertex V 0 1, v), we define 0 : r ^ Aut(S) as follows: 0(7) |v = 0(7), 0(y)(v) = v, and 0(7)(/ U {v}) = (0(y)(/)) U {v} for all f e F and 7 G r. Theorem 5.7. Lei S = (V, F; I) be a r-symmetric incidence structure (with respect to 9), and let S(r) be a (d — l)-picture which is T-symmetric with respect to 9 and t. Then the following hold: (i) S(r) has a non-trivial lifting if and only if the coned d-picture S(f), with the cone vertex at the point (0,..., 0, a) G Rd, for some non-zero a G R, has a non-trivial lifting. (ii) S is T-generically (minimally) flat (with respect to 9 and t ) in Rd-1 if and only if S is T-generically (minimally) flat (with respect to 9 and t) in Rd. Proof: (i) Let r1,..., r|V| be the vertices of the picture S(r). Embed S(r) into the space Rd via r = (r4,0) for i = 1,..., |V|. Then form the coned picture S(t), with the cone vertex r0 = (0,..., 0, a) G Rd, a = 0. The lifting matrix of S( r) is of the form M (S, r) (f ) The lifting matrix of the coned picture S(t) (with the cone vertex fixed) is of the form i fj m '(S,r) = (i,fj ) (0,/j ) 0 ... 0 1 0 ... 0 0 ... 0 [xi,yi,... ,Wi, 0,1] 0 ... 0 0 ... 0 0 0 ... 0 0 ... 0 [0, 0,..., 0, a, 1] 0 ... 0 Note that we added |F| rows (one for each incidence of the form (0, fj), j = 1,..., |F|, where 0 is the cone vertex) and |F| columns. Furthermore, for each added column (under face j) we have a 0 in each row, except in the one added row corresponding to the incidence (0, fj), where the entry is equal to a. Thus we have increased the rank by |F|, and preserved the dimension of the kernel. Now, if we add the missing column for the cone 44 Ars Math. Contemp. 13 (2017) 15-21 vertex, then we obtain the lifting matrix of the coned picture S(r ): 0 i _fj m (S ,r ) = (i,f) (0,fj) 0 1 0 ... 0 1 0 ... 0 0. . . 0 0 0. . . 0 0 ... 0 [xi,Vi,...,Wi, 0,1] 0 ... 0 0 ... 0 [0, 0,..., 0, a, 1] 0 ... 0 This added a 1-dimensional space of liftings (namely the space |Ard| A g R} of all 'vertical' translations of the picture (recall the proof of Prop. 5.4)), but did not add any non-trivial liftings. The rank of the matrix has not changed, nor has the space of row-dependencies. This proves (i). (ii) Note that if there exists some (minimally) flat r-symmetric realization of an incidence structure S, then clearly all T-generic realizations of S are also (minimally) flat. Therefore, by (i), it suffices to show that S(r) is r-symmetric with respect to 0 and t if and only if the coned picture S(r ) (with the cone vertex at the point (0,..., 0, a) g Rd, a = 0) is r-symmetric with respect to 0 and f. Let ri,..., r|V| be the vertices of the picture S(r), and let f0, fi,..., fV| be the vertices of the picture S(f ), i.e., f0 = (0,0,..., 0, a), and f = (rj, 0) for i = 1,..., | V|. For all y g r, we clearly have f (7)f0 = f0 = f<9(7)(o). Furthermore, for i = 0, S(r) is r-symmetric with respect to 0 and t if and only if f (Y)fi = (t 0) = (re(7)(i), 0) = (re(7)(i), 0) = r0(7)(i). This gives the result. □ (a) (b) (c) Figure 5: A C4-symmetric 2-picture S(r) (where all five interior regions are faces) (a) and a C4-symmetric coned picture S(f) in 3-space with cone vertex f0 = (0,..., 0, a) and rf = (r, 0) for i = 1,..., 8 (b). A C4-generic realization of S is shown in (c). Note that S also has five faces, namely the ones corresponding to the 'interior cells' of the cube in (c) except for the 'top cell' {f0, f 1, f2, f3, f4}. 0 1 V. E. Kaszanitzky and B. Schulze: Lifting symmetric pictures to polyhedral scenes 45 6 Further work In the previous sections we have derived new necessary conditions for a symmetric picture to be minimally flat. It is now natural to ask whether these conditions, together with the standard non-symmetric counts, are also sufficient for a picture which is realized generi-cally for the given symmetry group to be minimally flat. We conjecture that this is in fact the case, for all symmetry groups in all dimensions. Conjecture 6.1. A F-generic (d — 1)-picture S(r) is minimally flat if and only if (i) 111 = |V| + d|F| — d and \I'\ < |V'| + d|F'| — d for all nonempty subsets I' of incidences; (ii) S satisfies the conditions for F in the symmetry extended counting rule (Corollary 5.5); (iii) For every subset I' of I which induces a r'-symmetric incidence structure S' with |I'| = |V'| + d|F' | — d (where V Ç r), S' satisfies the conditions for r' in the symmetry extended counting rule. Note that if every face of the incidence structure S is incident to exactly four vertices (i.e., if S induces a 4-uniform hypergraph), then the count |I| = |V| +3|F| — 3 in condition (i) for d =3 simplifies to |F| = | V| — 3. Thus, a natural approach to prove this conjecture for d = 3 is to first focus on incidence structures which induce 4-uniform hypergraphs and to develop a symmetry-adapted version of the recently established recursive construction of 4-uniform (1,3)-tight hypergraphs [9]. For each of the symmetric hypergraph operations in this construction scheme, we then need to check that it preserves the full rank of the lifting matrix. Finally, one could then try to extend the result to general incidence structures with the same symmetry. Using this approach, Conjecture 6.1 has recently been verified for 2-pictures with three-fold rotational symmetry [11]. All the other cases, however, remain open, and we invite the reader to join in these explorations. Note that a similar approach based on symmetry-adapted recursive Henneberg-type graph constructions was successfully used in [16, 17] to establish symmetry-adapted versions of Laman's theorem for various groups. These results provide combinatorial characterizations of symmetry-generic isostatic (i.e. minimally infinitesimally rigid) graphs in the plane. However, the recursive construction of (non-symmetric) (1, 3)-tight hyper-graphs developed in [9] is more complex than the recursive Henneberg construction of (non-symmetric) Laman graphs, and hence Conjecture 6.1 presents us with new challenges of both combinatorial and geometric nature. For practical applications of scene analysis, it is of particular interest to develop methods which allow us to determine whether there exist a (unique) sharp lifting for a given picture, rather than just a non-trivial lifting. It is therefore natural to ask whether our results can be extended to provide necessary and/or sufficient conditions for a symmetric picture to be sharp, rather than just foldable. A combinatorial characterization of those pictures which have a unique sharp lifting if realized generically in (d — 1)-space (without symmetry) is given by the Picture Theorem (recall Theorem 2.1). This result is essentially a corollary of the combinatorial characterization of generic independent pictures (i.e., pictures whose lifting matrices have independent rows) given in [31]. Therefore, in order to obtain a complete symmetry-adpated version of the Picture Theorem we need to first obtain a combinatorial characterization of those 46 Ars Math. Contemp. 13 (2017) 15-21 pictures which are independent if realized generically with respect to the given symmetry group. Note that in the non-symmetric situation, any generic independent picture is a substructure of a minimally flat picture. In general, however, this is no longer the case for symmetric pictures. For example, it is easy to construct a C3-generic picture in the plane which satisfies |V3| = 1 and |131 = |F3| > 1, so that it is not contained in a minimally flat C3-generic picture (by Corollary 5.5), but whose lifting matrix has independent rows. It follows that a combinatorial characterization of T-generic minimally flat pictures would in general not provide us with a combinatorial characterization of T-generic independent pictures. However, once a characterization of T-generic independent pictures has been established for a group r (again using a symmetric recursive construction scheme, e.g.), then we expect that the proof idea in [31] can be extended to obtain a characterization of those pictures which have a unique sharp lifting if realized generically with respect to r. For some initial results for C3-generic pictures in the plane, we refer the reader to [11]. Acknowledgements We would like to thank Walter Whiteley for useful discussions. References [1] P. W. Atkins, M. S. Child and C. S. G. Phillips, Tables for Group Theory, OUP, Oxford, 1970. [2] R. Connelly, P. W. Fowler, S. D. Guest, B. Schulze and W. Whiteley, When is a symmetric pin-jointed framework isostatic?, International Journal of Solids and Structures 46 (2009), 762-773. [3] H. Crapo, Invariant-Theoretic Methods in Scene Analysis and Structural Mechanics, J. of Symbolic Computation 11 (1991), 523-548. [4] H. Crapo and W. Whiteley, Plane Self Stresses and Projected Polyhedra I: The Basic Pattern, Structural Topology 20 (1993), 55-78. [5] L. Cremona, (T. H. Beare, trans.), Graphical Statics, Clarendon Press, London, 1890. [6] P. W. Fowler and S. D. Guest, A symmetry extension of Maxwell's rule for rigidity of frames, International Journal of Solids and Structures 37 (2000), 1793-1804. [7] D. Huffman, Realizable Configurations of Lines in Pictures of Polyhedra, Machine Intelligence 8 (1977) 493-509. [8] G. James and M. Liebeck, Representations and Characters of Groups, Cambridge University Press, 1993. [9] T. Jordan and V.E. Kaszanitzky, Sparse hypergraphs with applications in combinatorial rigidity, Discrete Applied Math. 185 (2015), 93-101. [10] R. D. Kangwai and S. D. Guest, Symmetry-adapted equilibrium matrices, International Journal ofSolids and Structures 37 (2000), 1525-1548. [11] V. E. Kaszanitzky and B. Schulze, Characterizing minimally flat symmetric hypergraphs, EGRES Tech. Report TR-2015-17 (2015). [12] A. Mackworth, Interpreting Pictures of Polyhedral Scenes, Artificial Intelligence 4 (1973), 121-137. [13] J. C. Maxwell, On reciprocal figures and diagrams of forces, Phil. Mag. 27 (1864), Ser. 4, 250-261. V. E. Kaszanitzky and B. Schulze: Lifting symmetric pictures to polyhedral scenes 47 [14] L. Ros and F. Thomas, Geometric Methods for Shape Recovery from Line Drawings of Polyhedra, Journal of Mathematical Imaging and Vision 22 (2005), 5-18. [15] B. Schulze, Block-diagonalized rigidity matrices of symmetric frameworks and applications, Contributions to Algebra and Geometry 51 (2010), 427-466. [16] B. Schulze, Symmetric versions of Laman's Theorem, Discrete & Computational Geometry 44 (2010), 946-972. [17] B. Schulze, Symmetric Laman theorems for the groups C2 and Cs, The Electronic Journal of Combinatorics 17, R154, 1-61. [18] B. Schulze and S. Tanigawa, Infinitesimal rigidity of symmetric bar-joint frameworks, to appear in SIAMJ. Discrete Math, (arXiv:1308.6380), 2015. [19] B. Schulze and W. Whiteley, The Orbit Rigidity Matrix of a Symmetric Framework, Discrete & Comput. Geom. 46 (2011), 561-598. [20] B. Schulze and W. Whiteley, Coning, symmetry and spherical frameworks, Discrete & Comput. Geom. 48 (2012), 622-657. [21] B. Servatius and H. Servatius, The generalized Reye configuration, Ars Mathematica Contemporanea 3 (2010), 21-27. [22] K. Sugihara, Mathematical Structures of Line Drawings of Polyhedrons - Towards Man-Machine Communication by Means of Line Drawings, IEEE Trans. PAMI 4 (1982), 458-469. [23] K. Sugihara, An Algebraic and Combinatorial Approach to the Analysis of Line Drawings of Polyhedra, Discrete Appl. Math. 9 (1984), 77-104. [24] K. Sugihara, An Algebraic Approach to Shape-from-Image Problems, Artificial Intelligence 23 (1984), 59-95. [25] K. Sugihara, Machine interpretation of line drawings, MIT Press, Cambridge Mass, 1986. [26] N. White and W. Whiteley, The algebraic geometry of stresses in frameworks, SIAM J. Disc. Math. 4 (1993), 481-511. [27] W. Whiteley, Motions, stresses and projected polyhedra, Structural Topology 7 (1982), 13-38. [28] W. Whiteley, Cones, infinity and one story buildings, Structural Topology 8 (1983), 53-70. [29] W. Whiteley, A correspondence between scene analysis and motions of frameworks, Discrete Applied Math. 9 (1984), 269-295. [30] W. Whiteley, From a Line Drawing to a Polyhedron, J. Math. Psych. 31 (1987), 441-448. [31] W. Whiteley, A Matroid on Hypergraphs, with Applications in Scene Analysis and Geometry, Discrete & Comput. Geom. 4 (1989), 75-95. [32] W. Whiteley, Weavings, Sections and Projections of Spherical Polyhedra, Discrete Appl. Math. 32 (1991), 275-294. [33] W. Whiteley, Some Matroids from Discrete Applied Geometry, Contemporary Mathematics, AMS 197 (1996), 171-311. [34] W. Whiteley, Rigidity and Scene Analysis, Handbook of Discrete and Computational Geometry, J. E. Goodman, J. O'Rourke (eds), Chapman & Hall (2006), 1327-1354. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 49-61 On the Erdos-Sos Conjecture for graphs on n = k + 4 vertices* * Long-Tu Yuan, Xiao-Dong Zhang Department of Mathematics, and Ministry of Education, Key Laboratory of Scientific and Engineering Computing, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, P. R. China Received 1 August 2015, accepted 19 November 2015, published online 21 October 2016 The Erdos-Sos Conjecture states that if G is a simple graph of order n with average degree more than k — 2, then G contains every tree of order k. In this paper, we prove that Erdos-Sos Conjecture is true for n = k + 4. Keywords: Erdôs-S6s Conjecture, tree, maximum degree. Math. Subj. Class.: 05C05, 05C35 1 Introduction The graphs considered in this paper are finite, undirected, and simple (no loops or multiple edges). Let G = (V(G), E(G)) be a graph of order n, where V(G) is the vertex set and E (G) is the edge set with size e(G). The degree of v e V (G), the number of edges incident to v, is denoted dG(v) and the set of neighbors of v is denoted N(v). If u and v in V(G) are adjacent, we say that u hits v or v hits u. If u and v are not adjacent, we say that u misses v or v misses u. If S C V(G), the induced subgraph of G by S is denoted by G[S]. Denote by D(G) the diameter of G. In addition, ¿(G), A(G) and avedeg(G) = j/jf)| are denoted by the minimum, maximum and average degree in V(G), respectively. Let T be a tree on k vertices. If there exists an injection g : V(T) ^ V(G) such that g(u)g(v) e E(G) if uv e E(T) for u, v e V(T), we call g an embedding of T into G and G contains a copy of T as a subgraph, denoted by T C G. In addition, assume that T' C T is a subtree of T *This work is supported by National Natural Science Foundation of China (Nos.11271256 and 11531001), The Joint Israel-China Program (No.11561141001), Innovation Program of Shanghai Municipal Education Commission (No.14ZZ016) and Specialized Research Fund for the Doctoral Program of Higher Education (No.20130073110075). E-mail addresses: yuanlongtu@sjtu.edu.cn (Long-Tu Yuan), xiaodong@sjtu.edu.cn (Xiao-Dong Zhang) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract 50 Ars Math. Contemp. 13 (2017) 15-21 and g' is an embedding of T' into G. If there exists an embedding g : V(T) ^ V(G) such that g(v) = g'(v) for all v e V(T'), we say that g' is T-extensible. In 1959, ErdOs and Gallai [6] proved the following theorem. Theorem 1.1. Let G be a graph with avedeg(G) > k — 2. Then G contains a path of order k. Based on the above result, Later Erdos and Sos proposed the following well known conjecture (for example, see [5]). Conjecture 1.2. Let G be a graph with avedeg(G) > k — 2. Then G contains every tree on k vertices as a subgraph. Various specific cases of Conjecture 1.2 have already been proven. For example, Brandt and Dobson [2] proved the conjecture for graphs having girth at least 5. Balasubramanian and Dobson [1] proved this conjecture for graphs without any copy of K2,s, s < -k +1. Li, Liu and Wang [15] proved the conjecture for graphs whose complement has girth at least 5. Dobson [3] improved this to graphs whose complements do not contain K2,4. More results on this conjecture can be referred to [7, 8, 9] and [11, 12]. On the other hand, in 2003, Mclennan [10] proved the following theorem. Theorem 1.3. Let G be a graph with avedeg(G) > k — 2. Then G contains every tree of order k whose diameter does not excess 4 as a subgraph. In 2010, Eaton and Tiner [4] proved the the following two theorems. Theorem 1.4. [4] Let G be a graph with avedeg(G) > k — 2. If 5(G) > k — 4, then G contains every tree of order k as a subgraph. Theorem 1.5. [4] Let G be a graph with avedeg(G) > k — 2. If k < 8, then G contains every tree of order k as a subgraph. In 1984, Zhou [17] proved that Conjecture 1.2 holds for k = n. Later, Slater, Teo and Yap [13] and Wozniak [16] proved that Conjecture 1.2 holds for k = n — 1 and k = n — 2, respectively. Theorem 1.6. [16] Let G be a graph of order n with avedeg(G) > k — 2. If k = n — 2, then G contains every tree of order k as a subgraph. Recently, Tiner [14] proved that Conjecture 1.2 holds for k = n — 3. Theorem 1.7. [14] Let G be a graph of order n with avedeg(G) > k — 2. If k > n — 3, then G contains every tree of order k as a subgraph. In this paper, we establish the following: Theorem 1.8. Let G be a graph of order n with avedeg(G) > k — 2. If k > n — 4, then G contains every tree of order k as a subgraph. L.-T. Yuan andX.-D. Zhang: On the Erdos-Sos Conjecture for graphs on n = k + 4 vertices 51 2 Proof of Theorem 1.8 Let T be any tree of order k. If k > n — 3, or k < 8 or the diameter of T is at most 4, the assertion holds by Theorems 1.3, 1.5 and 1.7. We only consider k = n — 4 > 9, D(T) > 5 and prove the assertion by the induction. Clearly the assertion holds for n = 6. Hence assume Theorem 1.8 holds for all of the graphs of order fewer than n and let G be a graph of order n. If there exists a vertex v with dG(v) < |_f J, then avedeg(G — v) > k — 2 and the assertion holds by Theorems 1.7. Furthermore, by Theorem 1.4, without loss of generality, there exists a vertex z in V (G) such that |_ f J < dG (z ) = ¿(G) < k —5. Without loss of generality, we can assume that e(G) = 1 + |_ 2 (k — 2)(k + 4) J. Let T be any tree of order k with a longest path P = a0ai... ar_iar and NT(ai) \ {a2} = {&i,..., bs} and Nt(ar-1) \ {ar-2} = {c1,..., ct}. Since avedeg(G) > k — 2, we can consider the following cases: A(G) = k + 3, k + 2, k + 1, k, k — 1. 2.1 A(G) = k + 3 Let u G V(G) be such vertex that dG(u) = k + 3 and let G' = G — {u, z} and T' = T — {a1, b1,..., bs}. Then e(G') > e(G) — (k + 3) — (k — 5) + 1 > 2(k + 4)(k — 2) — (k + 3) — (k — 5) +1 = 1 (k2 — 2k — 2). So avedeg(G') > (k2 — 2k — 2)/(k + 2) > k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. Let f' be an embedding of T' into G'. Then let f = f' in T' and f (a1) = u, X = V(G) \ f'(V(T')). Since dG(u) = k + 3, u hits at least s vertices in X. Hence f can be extended to an embedding of T into G or we can say that f is T—extensible. Remark: For the remainder of this paper we shall always let f' be an embedding of T' into G' and when we do not define the value of f on any vertex of T', we always let f = f' on those vertices. 2.2 A (G) = k + 2 Let u G V(G) be such vertex that dG (u) = k + 2. Then there exists only one vertex x G V(G) \ {u} not adjacent to u. We consider two subcases: dG(x) < k — 2 and dG(x) > k — 1. 2.2.1 dG(x) < k - 2 Let G' = G — {u, x} andT' = T — {a1, b1,..., bs}. Then e(G') > e(G) — (k + 2) — (k — 2) > 1 (k2 —2k — 8). Soavedeg(G') > (k2 —2k — 8)/(k+2) = k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. Then let f (a1) = u and X = V(G) \ f'(v(T')). Since dG(u) = k + 2, u hits at least s vertices in X, f is T—extensible. 2.2.2 dG(x) > k - 1 Since x = z, we consider the following two cases. (A). x misses z. Let G' = G — {u, z, x} and T' = T — {a1, b1,..., bs, ar}. Then e(G') > e(G) — (k + 2) — (k — 5) — (k + 1) + 1 > 2(k2 — 4k — 2). Hence avedeg(G') > (k2 — 4k — 2)/(k + 1) > k — 5 and | V(T') |< k — 3. By the induction hypothesis, we have T' C G'. Since x misses z, u and dG(x) > k — 1, x misses at most two vertices of G'. If x hits f'(a2), let f (a1) = x and f (ar) = u. Since dG(x) > k — 1 and u hits all vertices of T', f is T—extensible. Hence we assume that x misses f'(a2). If x hits f'(ar-1), let 52 Ars Math. Contemp. 13 (2017) 15-21 f (ar) = x and f (ai) = u. Then f is T-extensible. If x misses f '(a2) and f '(ar-i), then x hits all of V(G') \ {f '(a2), f '(ar-i)}, because D(T) > 5, a2 and ar-i are not adjacent. Then let f (ar-i) = x, f (ai) = u, which implies that f is T-extensible. (B). x hits z. We consider the following two subcases. (B.1). dG(x) > k - 1. Let G' = G - {u,z,x}, T' = T - {ai, bi,..., bs, ar}. Since x misses u and dG(x) > k - 1, x misses at most two vertices of G', the assertion can be proven by the similar method of (A). (B.2). dG(x) = k - 1. Then x misses 3 vertices of V(G) \ {u}, says yi, y2, y3. (a). There exists one vertex yj with 1 < i < 3 such that dG(yj) > k + 1. Let G' = G - {u, z, yj, x} and T' = T - {ai, bi,..., 6s, ar-i, ci,..., ct}. Then e(G') > e(G) - (k + 2) - (k - 5) - (k + 2) - (k - 1) + 3 + 1 > ± (k2 - 6k + 4), which implies avedeg(G') > (k2 - 6k + 4)/k > k - 6 and | V(T') |< k - 4. Hence by the induction hypothesis, T' C G'. Note that yj misses at most one vertex of G'. If yj misses f'(a2), let f (ai) = u, f (ar_i) = yj; if yj misses f'(ar_2), let f (ar_i) = u,f (ai) = yj. Thus f is T-extensible. (b). There exists one vertex yj with 1 < i < 3 such that dG(yj) = k and yj misses z. Then the proof is similar to (a) and omitted. (c). There exists one vertex yj with 1 < i < 3 such that dG(yj) < k - 2. Let G' = G-{u, yj, x} andT' = T-{ai, bi,..., bs, ar}. Then e(G') > e(G)-(k+2)-(k-2)-(k-1) + 1 > 2(k2-4k-4), which implies avedeg(G') > (k2-4k-4)/(k+1) > k-5 and | V(T') |< k - 3. Hence by the induction hypothesis, T' C G'. Similarly as in case (A), there exists an embedding from T into G. (d). dG(yj) = k, yj hits z or dG(yj) = k - 1 for i G {1,2,3}. (d.1). dT(ai) + dT(ar-i) > 5. Let G' = G - {u, z, yi, y2, x} and T' = T -{ai, bi,..., bs, ar-i, ci,..., ct}. Then e(G') > e(G)-(k+2)-(k-5)-(k-1)-(k-1)-(k -1) + 3 > 2 (k2-8k +10) which implies avedeg(G') > (k2 - 8k + 10)/(k -1) > k - 7 and | V(T') |< k - 5. Hence by the induction hypothesis, T' C G'. Moreover, x misses only one vertex of G'. If x misses f'(a2), let f (ai) = u, f (ar-i) = x; if x misses f '(ar-2), let f (ar-i) = u, f (ai) = x. In both situations, f is T-extensible. (d.2). dT(ai) = dT(ar-i) = 2. Let G' = G - {u, z} and T' = T - {a0, ai}. Then e(G') > e(G) - (k + 2) - (k - 5) + 1 > ± (k2 - 2k), which implies avedeg(G') > (k2 - 2k)/(k + 2) > k - 4 and | V(T') |< k - 2. By the induction hypothesis, T' C G'. Moreover, u hits all vertices of V(G) \ {x} and z hits x. Let f (ai) = u or z and f (a0) = z or u. Then f is T-extensible. 2.3 A(G) = fc +1 Let u G V(G) be such vertex that dG(u) = k + 1 with u missing vertices xi and x2. Without loss of the generality, we can assume dG(xi) > dG(x2) and (ai) > (ar-i). 2.3.1 dT(ai) + dT(ar-1) > 5 We consider the following two cases: (A) and (B). (A). xi misses x2. (A.1) dG(xi) + dG(x2) < 2k - 3. Let G' = G - {u,xi,x2} and T' = T -{ai, bi,..., bs}. Then e(G') > e(G) - (k +1) - (2k - 3) > i(k2 - 4k - 4), which implies avedeg(G') > (k2 - 4k - 4)/(k + 1) > k - 5 and | V(T') |< k - 3. Hence by the induction hypothesis, T' C G'. Let f (ai) = u. It is easy to see that f is T-extensible. L.-T. Yuan andX.-D. Zhang: On the Erdos-Sos Conjecture for graphs on n = k + 4 vertices 53 (A.2). dG(x1) + dG(X2) > 2k — 2. (a). dG(x1) = k — 1 Then dG(x2) = k — 1 andx1 misses {u,x2,y1,y2}. If y1,y2 = z, let G' = G — {u, z, x1,x2,y1} and T' = T — {a1,b1,... ,bs ar-1, c1,..., ct}. Then e(G') > e(G) — (k + 1) — (k — 5) — (2k — 2) — (k + 1) + 3 > ±{k2 — 8k + 8), which implies avedeg(G') > (k2 — 8k + 8)/(k — 1) > k — 7 and | V(T') |< k — 5. Hence by the induction hypothesis, T' C G'. Note that x1 misses only one vertex of G'. If x1 misses f '(a2), let f (a1) = u and f (ar-1) = x1; if x1 misses f '(ar-2), let f (ar-1) = u and f (a1) = x1. In both situations, f is T—extensible. Now assume that y1 = z or y2 = z. Let G' = G — {u, x1,x2 ,y1,y2} and T' = T — {a1,b1,..., bs,ar-1,c1,..., ct}. Then e(G') > e(G) — (k +1) — (k — 5) — (2k — 2) — (k + 1) + 2 + 1 > ± (k2 — 8k + 8), which implies avedeg(G') > (k2 — 8k + 8)/(k — 1) > k — 7 and | V(T') |< k — 5. Let f (ar-1) = u and f (a1) = x1. Then f is T—extensible. (b). da(x1) > k. Let G' = G — {u, z, x1, x2} and T' = T — {a1, b1,... ,bs, ar-1, c1, ..., ct}. Then e(G') > e(G) — (k + 1) — (k — 5) — (2k + 2)+ 1 + 2 > 2 (k2 — 6k + 2), which implies avedeg(G') > (k2 — 6k + 2)/k> k — 6 and | V(T') < k — 4. Hence by the induction hypothesis, T' C G'. Note that x1 misses at most one vertex of G'. If x1 misses f'(a2), let f (a1) = u and f (ar-1) = x1; if x1 misses f'(ar-2), let f (ar-1) = u and f (a1) = x1. In both situations, f is T—extensible. (B). x1 hits x2. (B.1). dG(x1) + dG(x2) < 2k — 2. The proof is similar to (A.1) and omitted. (B.2). dG(x1) + dG(x2) > 2k — 1. The proof is similar to (A.2) with (a)dG(x^) = k,dG(x2) = k — 1 or k, (b)dG(x1) = k +1. 2.3.2 dT (ai) = dT (ar-1) = 2. We consider the following four cases. (A). There exists a vertex v = u of degree at most k such that it hits both x1 and x2. Let G' = G — {u, v} and T' = T — {a0, a1}. Then e(G') > e(G) — (k + 1) — k +1 > 1 (k2 — 2k — 8), which implies avedeg(G') > (k2 — 2k — 8)/(k + 2) = k — 4 and i V(T') l< k — 2. Hence by the induction hypothesis, T' C G'. If f '(a2) hits u, let f (a1) = u. If f'(a2) misses u, then f '(a2) = x1 or x2 and let f (a1) = v, f (a0) = u. Thus f is T—extensible. (B). There exists a vertex v = u of degree at least k + 1 such that it hits both x1 and x2. Then dG(v) = k + 1 and v misses y1 and y2. Since the case z G {x-]_, x2,y1, y2} is much easier, we may suppose z = x1 ,x2, y1,y2. Let G' = G — {u, v, z} — x1x2 — y1y2 and T' = T — {a0,a1,ar}. Then e(G') > e(G) — 2(k + 1) — (k — 5) + 1 — 2 > 2(k2 — 4k — 4), which implies avedeg(G') > (k2 — 4k — 4)/(k + 1) > k — 5 and | V(T') l< k — 3. Hence by the induction hypothesis, T' C G'. If f '(a2) = x1 or x2, and f '(ar-1) = y1 or y2, then let f (a1) = v and f (ar) = u. If f '(a2) = x1 and f '(ar-1) = x2, then let f (a1) = v,f (ar-1) = u, because u hits all the neighbours of f'(ar-1). If f '(a2) = y1,f '(ar-1) = y2, then let f (a1) = u and f (ar-1) = v. For the rest situations, it is easy to find an embedding from T into G. (C). There is no vertex in V(G) \ {u} hitting both x1 and x2, and x1 misses x2. Then dG(x1) + dG(x2) < k + 1. Let G' = G — {u, x1, x2} and T' = T — {a0, a1}. Then e(G') > e(G) — (k + 1) — (k + 1) > 2(k2 — 2k — 12), Since k > 9, avedeg(G') > (k2 — 2k — 12)/(k +1) > k — 4 and | V2(T') l< k — 2. By theorem 1.7, T' C G'. Let f (a1) = u. Then f is T—extensible. 54 Ars Math. Contemp. 13 (2017) 15-21 (D). There is no vertex in V(G) \ {u} hitting both x1 and x2, and x1 hits x2. Then dG (x1) + dG(x2) < k + 3. If dG(x1) + dG(x2) < k + 2, the assertion follows from (C). Hence assume that dG(x1) + dG(x2) = k + 3. If z = x1, x2, then z has to hit x1 or x2, say that z hits x1. Let G' = G — {u, z} — x1x2 and T' = T — {ao, a1}. Then e(G') > e(G) — (k +1) — (k — 5) + 1 — 1 > 1 (k2 — 2k), which implies avedeg(G') > (k2 — 2k)/(k + 2) > k — 4 and | V(T') |< k — 2. Hence by the induction hypothesis, T' C G'. If f '(a2) hits u, let f (a1) = u; if f '(a2) = x1, let f (a1) = z and f (a0) = u. If f'(a2) = x2 and if there is a vertex w in T' such that f'(w) = x1, let f (w) = u, f (a1) = x1 and f (a0) = z, because u hits all neighbours of f'(w) in G'; if f'(a2) = x2 and there does not exist any vertex w in T' such that f '(w) = x1, let f (a1) = x1, and f (a0) = z. In all situations, f is T—extensible. If z = x1 or x2, then let G' = G — {u, z} and T' = T — {a0, a1}. Similarly, we can find an embedding from T into G. 2.4 A(G) = fc Let u G V(G) be a vertex of degree dG(u) = k and misses three vertices x1,x2,x3. Denote by S = {x1, x2, x3}. 2.4.1 G[S] contains no edges. Let G' = G — {u} and T' = T — {a0}. Then e(G') > e(G) — k > 1 (k2 — 8), which implies avedeg(G') > (k2 — 8)/(k + 3) > k — 3 and | V(T') |< k — 1. By the induction hypothesis, T' C G'. If f '(a1) hits u, let f (a0) = u; if f '(a1) = x,, 1 < i < 3, let f (a1) = u. Since u hits all neighbours of f '(a1) in G', f is T—extensible. 2.4.2 G[S] contains exactly one edge. Without loss of the generality, x1 hits x2, dG(x1) > dG(x2), and dT (a1) > dT (ar-1). We consider two cases. (A). dx(a1) + dT(ar-1) > 5. (A.1). dG(x2) > k — 1. If x3 = z, let G' = G — {u, z, x3} — x1x2 and T' = T — {a1, b1,..., bs}. Then e(G') > e(G) — k — (k — 5) — k — 1 > 2(k2 — 4k), which implies avedeg(G') > (k2 — 4k)/(k + 1) > k — 5 and | V(T') |< k — 3. By the induction hypothesis, T' C G'. If f'(a2) hits u, then let f (a1) = u; if f'(a2) = x1 and x2 G f'(V(T')), then let f (a^ = x2; if f'(a2) = x1 and x2 G f'(V(T')) and f'(w) = x2, then let f (w) = u, f (a2) = x1, and f (a1) = x2. Hence f is T—extensible. On the other hand, if x3 = z, let G' = G — {u, z} — {x1x2} and T' = T — {a1, b1,..., bs}. Similarly, we can prove that the assertion holds. (A.2). dG(x3) > k — 1. By (A.1), we can assume that dG(x2) < k — 2. If z = x1, x2, let G' = G — {u, z, x1, x2, x3} and T' = T — {a1, b1,..., bs, ar-1, c1,..., ct}. Then e(G') > e(G) — k — (k — 5) — (k — 2) — k — k + 2+1 > 2(k2 — 8k + 12), which implies avedeg(G') > (k2 — 8k + 12)/(k — 1) > k — 7 and | V(T') |< k — 5. Hence by the induction hypothesis, T' C G'. Moreover, x3 misses at most one vertex of V(G'). If x3 misses f '(a2), let f (a1) = u and f (ar-1) = x3; if x3 hits f '(a2), let f (ar-1) = u and f (a1) = x3. then f is T—extensible. On the other hand, if x1 = z or x2 = z, let G' = G — {u, x1, x2, x3} and T' = T — {a1, b1,..., bs, ar-1, c1,..., ct}. Using the same above argument, we can prove the assertion. L.-T. Yuan andX.-D. Zhang: On the Erdos-Sos Conjecture for graphs on n = k + 4 vertices 55 (A.3). dG(xi) = k,dG(x2) < k — 2 and dG(x3) < k — 2. If z = x2,x3, let G' = G — {u, z, x1; x2, x3} and T' = T — ja1, b1;..., 6s, ar_1; c1,..., ct}. Hence e(G') > e(G) — k — (k — 5) — (k — 2) — k — (k —2)+2 > 1 (k2 —8k +10), which implies avedeg(G') > (k2 — 8k + 10)/(k — 1) > k — 7 and | V(T') |< k — 5. By the induction hypothesis, T' C G'. Note that x1 misses at most one vertex in V(G'). If x1 misses /'(a2), let f (a1) = u and /(ar-1) = x1; if x1 hits /'(a2), let /(ar-1) = u and /(a1) = x1. Hence / is T—extensible. On the other hand, if x2 = z or x3 = z, let G' = G — {u, x1, x2, x3} and T' = T — {a1, b1,..., 6s, ar-1, c1,..., ct}. By the same above argument, we can prove the assertion. (A.4). dG(x1) < k — 1,dG (x2) < k — 2 and dG(x3) < k — 2. Then there exists a vertex u' in V(G) \ {x1, x2, x3, u} with degree at least k — 1. Otherwise, by ¿(G) < k — 5, we have avedeg(G) < k+(k_1)(k_2)+(k_1)+2(fc_2)+(fc_5) < k — 2, which is a contradiction. Let G' = G — {u, u'} — {x1x2} and ++' = T — {fl1, b1,..., bs}. Then e(G') > e(G) — k — k + 1 — 1 > 1 (k2 — 2k — 8), which implies avedeg(G') > (k2 — 2k — 8)/(k + 2) = k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. If /'(a2) hits u, let /(a1) = u; if /'(a2) misses u, let /(a2) = u and /(a1) = u'. Then / is T—extensible. (B). dT (a1) = 2 and dT (ar_1) = 2. If there exists a vertex w that hits both x1 and x3, let G' = G — {u, w} — x1x2 and T' = T — {a0, a1}. Then e(G') > e(G) — 2k + 1 — 1 > 1 (k2 — 2k — 8), which implies avedeg(G') > (k2 — 2k + 8)/(k + 2) = k — 4 and i V(T') |< k — 2. By the induction hypothesis, T' C G'. If /'(a2) = x1 or x3, let /(a1) = w and /(a0) = u; if /'(a2) = x2 and x1 G /'(V(T')), let /(a1) = x1 and /(ao) = w; if /'(«2) = x2 and x1 G /'(V(T')),/'(v) = x1, let /(v) = u, /(«1) = x1 and /(a0) = w. In the above situations, / is T—extensible. On the other hand, if there is no vertex hits both x1 and x3,or x2 and x3. then dG(x1)+dG(x3) < k, dG(x2)+dG(x3) < k. Since dG(xj) > [|J and k > 9, dG(xj) < k — 2. Hence, similarly as in (A.4), there exists a vertex u' in V(G) \ {x1, x2, x3, u} with degree at least k — 1, and an embedding of T into G. 2.4.3 G[S] contains exactly two edges Without loss of the generality, assume that x1 hits both x2 and x3. We consider the following two cases. (A). dT(a1) = 2. Let G' = G — {u,x1} and T' = T — {a0,a1}. Then e(G') > e(G) — 2k > 1 (k2 — 2k — 8), which implies avedeg(G') > (k2 — 2k — 8)/(k + 2) > k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. If /'(«2) = x2 or x3 (say x2), let /(«1) = x1; Moreover, if x3 / /'(V(T')), let /(«0) = x3; if x3 G /'(V(T')) and /'(v) = x3, let /(v) = u, /(«1) = x1, and /(«0) = x3. Hence, / is T-extensible. If /'(«2) = x2, x3, then it is easy to find an embedding from T to G. (B). dT(«1) > 3. (a). dG(x1) > k — 1. If z = x2,x3, let G' = G — {u,z,x1} and T' = T — {«1, b1,..., bs}. Then e(G') > e(G) — k — (k — 5) — k +1 > 2(k2 — 4k + 4), which implies avedeg(G') > (k2 — 4k + 4)/(k +1) > k — 5 and | V(T') |< k — 3. By the induction hypothesis, T' C G'. If /'(«2) = x2 or x3 (say x2), let /(«1) = x1. Moreover, if x3 G /'(V(T')), let /(61) = x3; if x3 G /'(V(T')) and /'(v) = x3, let /(v) = u, /(«1) = x1 and /(61) = x3. Because u hits all neighbours of /'(v) and dG(x1) > k — 1, / is T—extensible. If /'(«2) = x2, x3, it is easy to find an embedding from T to G. On the other hand, if z = x2 or x3 (say x2), let G' = G — {u, x1, x2}, by the 56 Ars Math. Contemp. 13 (2017) 15-21 same argument above, the assertion holds. (b). dG(xi) < k — 2, dG(x2) = k or dG(x3) = k (say dG(x2) = k). Then there exists a vertex y G V(G) \ {u, x1,x2,x3} such that x2 misses y. So x2 misses u, x3 and y and u misses x3. By Case 2.4.2, we can assume y hits x3. Further, by (a), we can assume dG(y) < k — 2. If z = x1,y, let G' = G — {u, z, x2, x3, y} and T' = T — {a1, b1,..., 6s, or-1, c1,..., ct}. Then e(G') > e(G) — k — (k — 5) — k — k — (k — 2) + 3 > 2(k2 — 8k + 12), which implies avedeg(G') > (k2 — 8k + 12)/(k — 1) > k — 7 and | V(T2') |< k — 5. By the induction hypothesis, T' C G'. Further, if f '(02) = X1, let f (a1) = x2 and f (or-1) = u; if f'(or_2) = x1, let f (or-1) = x2 and f (a1) = u. Hence f is T—extensible. On the other hand, if z = y, let G' = G — {u, x2, x3, y} and T' = T — {01,61,..., bs, ar_1, C1,..., ct}; if z = x1, let G' = G — {u, z, x2, x3, y} and T' = T — {a1, 61,..., 6s, ar-1, c1,..., ct}. Then by the same argument, it is easy to prove that the assertion holds. (c). dG(x1) < k — 2, dG(x2) = k — 1 and dG(x3) = k — 1. Let G' = G — {u, x2, x3} andT' = T — {04,61,..., 6s}. Then e(G') > e(G)—k — (k — 1) —(k — 1) > 1(k2 —4k—4), which implies avedeg(G') > (k2 — 4k — 4)/(k +1) > k — 5 and | V(T') |< k — 3. Bythe induction hypothesis, T' C G'. If f'(a2) = x1, let f (o1) = x2, which f is T—extensible. If f '(a2) = x1, it is easy to find an embedding from T to G. (d). dG(x1) < k — 2, and dG(x2) < k — 2 or dG(x3) < k — 2 (say dG(x2) < k — 2), hence dG(x3) < k — 1 by (b). Then there exists a vertex u' G V(G) \ {x1,x2,x3,u} of degree at least k — 1, otherwise 2e(G) < (k — 1)(k — 2) + (k — 5) + k + 2(k — 2) + (k — 1) < (k + 4)(k — 2) which is impossible. Let G' = G — {u,u',x1} and T' = T — {a1, 61,..., 6s}. Then e(G') > e(G) — 2k — (k — 2) + 1 > 1 (k2 — 4k — 2), which implies avedeg(G') > (k2 — 4k — 2)/(k +1) > k — 5 and | V (T') |< k — 3. Bythe induction hypothesis, T' C G'. Hence if f'(o2) hits u, let f (o1) = u; if f'(a2) = x2 or x3 (say x2), let f (o2) = u and f (o1) = u' since u hits all the neighbours of f'(o2). Then f is T—extensible. 2.4.4 G[S] contains exactly three edges The following two cases are considered. (A). dT (a1) = 2. If there exists an 1 < i < 3 (say i = 1) such that dG(x1) < k — 1, let G' = G — {u,x1}— x2x3 and T' = T — {a0, a1}. Then e(G') > e(G) — k — (k — 1) — 1 > 1 (k2 — 2k — 8), which implies avedeg(G') > (k2 — 2k — 8)/(k + 2) = k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. If f'(o2) = x2 or x3 (say x2), let f (o1) = x1. Moreover, if x3 G f'(V(T')), let f (00) = x3; and if x3 G f'(V(T')) and f'(v) = x3, let f (v) = u, f (a1) = x1, f (a0) = x3. Hence f is T—extensible. On the other hand, if dG(x1) = dG(x2) = dG(x3) = k, let G' = G — {u, x1} and T' = T — {a0, a1}. Then e(G') > e(G)—2k > 2 (k2 —2k — 8), which implies avedeg(G') > (k2 —2k — 8)/(k+2) = k — 4 and | V(T') |< k — 2. By the induction hypothesis,T' C G'. If f'(a2) = x2 or x3, let f (a1) = x1; if f'(a2) = x2, x3, let f (a1) = u. Hence f is T-extensible. (B). dT(a1) > 3. If there exists an 1 < i < 3 (say i = 1) such that dG(x1) > k — 1, Let G' = G — {u,z,x1}— x2x3. By the same argument as Case 2.4.3.(B).(a), the assertion holds. The rest is similar as Case 2.4.3.(B).(d). L.-T. Yuan andX.-D. Zhang: On the Erdos-Sos Conjecture for graphs on n = k + 4 vertices 57 2.5 A (G) = k - 1 Since A(G) = k - 1 and ¿(G) < k - 5, there exist at least four vertices of degree k - 1. Otherwise 2e(G) < 3(k-1)+k(k-2) + (k - 5) = (k-2)(k+4), which is a contradiction. Let ui be vertex of da(ui) = k - 1 missing four vertices of Si = {x^, xi2, xi3, xi4} for i = 1,2, 3,4. If there exists a vertex ui with 1 < i < 4 such that G[Si] contains at most one edge. Let G' = G - {uj} - E(G[Sj]) and T' = T - {ao}. Then e(G') > e(G) - (k - 1) - 1 > i(k2 - 8), which implies avedeg(G') > (k2 - 8)/(k + 3) > k - 3 and | V(T') |< k - 1. By the induction hypothesis, T' Ç G'. If ui hits f '(a1), let f (a0) = ui, and if ui misses f '(a1 ), let f (a1) = ui. Then f is T-extensible. Hence we assume that G[Si] contains at least two edges for i = 1, 2, 3,4. 2.5.1 dT(at) > 3, dT(ar-1) > 2 We consider the number of the edges in G[«i, u2, «3, «4]. (A). G[u1, u2, «3, u4] contains at least one edge, say u1 hits u2. If z G = {x11, x12, x13,x14}, let G' = G - {u1,u2,z} - E(G[S1]) and T' = T - {a1, b1,..., bs}. Then e(G') > e(G) - 2(k -1) - (k - 5) + 1 - 6 > ± (k2 - 4k - 4), which implies avedeg(G') > (k2 - 4k - 4)/(k +1) > k - 5 and | V(T ) |< k - 3. By the induction hypothesis, T' C G'. Hence if u1 hits f'(a2), let f (a1) = u1; and if u1 misses f '(a2), let f (a2) = u1 and f (a1) = u2. Since u1 hits all the neighbours of f '(a2) in G', f is T-extensible. On the other hand, if z G S1 = {x11, x12, x13, x14}, say z = x11. Let G' = G - {u1, u2, z} -E(G[x12, x13, x13]). By the same argument, the assertion holds. (B). G[u1, u2, u3, u4] contains no edges. (B.1). If there exist two vertices, say u1 and u2, in {u1, u2, u3, u4} such that u1 misses y1 and u2 misses y2, where y1 = y2 and y1,y2 G {« , ...,«4}. Let G' = G - {«1, «2, «3, «4} and T' = T - {01, 61,..., 6s, ar_1, C1,..., ct}. Then e(G') > e(G) - 4(k - 1) > 1 (k2 - 6k), which implies avedeg(G') > (k2 - 6k)/k = k - 6 and | V(T') |< k - 4. By the induction hypothesis, T' C G'. Hence if f'(«2) = y1, let f (01) = «2 and f (ar_1) = «1; if f '(02) = y2, let f (01) = «1 and f (ar_1) = «2. Moreover, if f '(ar_2) = y1, let f (a1) = «1 and f (ar-1) = «2; and if f '(ar_2) = y2, let f (a1) = «2 and f (ar-1) = «1. Therefore, f is T-extensible. (B.2). There exist a vertex y G {«1,..., «4} such that y misses «1,..., «4. Then G[«1,..., «4, y] contains no edges. (a). dT(ar-1) = 2. Then there exists a vertex w hits {«1, «2, «3, w4} and y. Let G' = G-{«1,w} andT' = T-{ar-1,ar}. Then e(G') > e(G)-2(k-1) + 1 > 2(k2-2k-2), which implies avedeg(G') > (k2 - 2k - 2)/(k + 2) > k - 4 and | V(T') |< k - 2. By the induction hypothesis, T' C G'. Hence if f '(ar_2) = «2, «3, «4 or y, let f (ar-1) = w and f (ar) = «1; and if f'(ar_2) = «2, «3, «4, y, let f (ar_1) = «1 and f (ar) = w. Therefore f is T-extensible. (b). dT(ar_1) > 3. If z = y, let G' = G - {«1, «2, «3, «4, y, z} and T' = T -{a1,61, ...,6s,ar_1,c1,.. .,ct}. Then e(G') > e(G) - 4(k - 1) - (k - 1) - (k - 5)+4 > 1 (k2 - 10k + 20), which implies avedeg(G') > (k2 - 10k + 20)/(k - 2) > k - 8 and i V(T') |< k - 6. By the induction hypothesis, T' C G'. Let f (01) = «1 and f (ar _ 1) = «2. Then f is T-extensible. On the other hand, if z = y, let G' = G - {«1, «2, «3, «4, z} and T' = T-{a1, 61,..., 6s, ar _ 1, c1,..., ct}. By the same argument, the assertion holds. 58 Ars Math. Contemp. 13 (2017) 107-123 2.5.2 dT(a1) = 2, dT(ar-1) = 2. We will discuss the following four cases: (A), (B), (C) and (D). (A). There exists a 1 < i < 4, say i = 1, such that G[S1] contains two or three edges. If mi hits one vertex, say u2, of three vertices u2, u3, u4. Let G' = G — {u1, u2} — E(G[S1]) and T' = T — {a0, a1}. Then e(G') > e(G) — 2(k — 1) + 1 — 3 > 2(k2 — 2k — 8), which implies avedeg(G') > (k2 — 2k — 8)/(k + 2) = k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. Hence if u1 hits /'(a2), let / (a1) = u1; and if u1 misses / '(a2), let /(a2) = u1 and /(a1) = u2. Since u1 hits all the neighbours of /'(a2) in G', / is T—extensible. Therefore, we assume that u1 misses uj for j = 2, 3,4. Then u1 misses x11 = u2,x12 = u3, x13 = u4, x14 and G[u2, u3, u4, x14] contains two or three edges. (A.1). x14 hits one vertex, say u2, of three vertices u2,u3,u4. Let G' = G — {u1, u2,u3,u4} and T' = T — {a0, a1, ar-1, ar}. Then e(G') > e(G) — 4(k — 1) > 1 (k2 — 6k), which implies avedeg(G') > (k2 — 6k)/k = k — 6 and | V(T') |< k — 4. By the induction hypothesis, T' C G'. Since G[m2, u3, u4, x14] contains two or three edges, there exists a vertex, say u3, of two vertices u3,u4 misses at most one vertex, say y1, in V(G) \ {u1,u2,u4, x14}. Hence if /'(a2) = x14 or y1, and /'(ar-2) = y1 or x14, let /(a1) = u2 or u1 and /(ar-1) = u1 or u2, then / is T—extensible. For the rest cases, it is easy to find an embedding from T to G. (A.2). x14 misses three vertices u2, u3, u4. Then G[u2, u3, u4] contains two or three edges. We can assume that u2 hits u3 and u4. If u3 misses u4, u3 misses at most one vertex, says y1, in V(G) \ {u1, u2, u4, x14}. Then let G' = G — {u1, x14, u3, u4} and T' = T — {a0, a1, ar-1, ar}. By the similar argument as Case (A.1), the assertion holds. Hence we can assume that u3 hits u4 and u3 misses z1, z2, u1, x14. Let G' = G — {u1, x14, u3, u4} — {z1z2} and T' = T — {a0, a1, ar-1, ar}. Then e(G') > e(G) — 4(k — 1) + 1 — 1 > 1 (k2 — 6k), which implies avedeg(G') > (k2 — 6k)/k = k — 6 and | V(T') |< k — 4. By the induction hypothesis, T' C G'. Hence if /'(a2) = z1 or z2, and /'(ar-2) = z2 or z1, let /(a2) = u3, /(a1) = u4, /(ar-1) = u1. Therefore / is T — extensible. If /'(a2) = z1 or z2, and /'(ar-2) = u2, let /(a1) = u1, /(ar-1) = u4. Therefore / is T — extensible. For the rest cases, it is easy to find an embedding from T to G. (B). There exists a 1 < i < 4, say i =1, such that G[S1] contains exactly four edges. (B.1). There exists a vertex, say x11, of degree 3 in G[S1] and | E(G[S1]) |< 5. Then x11 hits x12,x13 and x14. Let G' = G — {u1,x11} — E(G[x12, x13,x14]) and T' = T — {a0, a1}. Then e(G') > e(G) — 2(k — 1) — 2 > 2(k2 — 2k — 8), which implies avedeg(G') > (k2 — 2k — 8)/(k + 2) = k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. Hence if u1 hits /'(a2), let /(a1) = u1, which implies that / is T—extensible. If u1 misses /'(a2) and /'(a2) = x12, let /(a1) = x11. Moreover, if x13 or x14 ^ /'(V(T')), then let /(a0) = x13 or x14. Then / is T—extensible. If x13 and x14 € /'(V(T')), /'(w) = x13 or x14, let /(w) = u1, /(a0) = x13 or x14. Then / is T—extensible. For the rest cases, it is easy to find an embedding from T to G. (B.2). The degree of every vertex in G[S1] is two. We assume that x11 hits x12, x12 hits x13, x13 hits x14, x14 hits x11. (a). u1 hits all vertices of {u2, u3, u4}. (a.1). There exists a vertex uj, say u2, in {u2, u3, u4} which misses x11, x12, x13 and x14. Let G' = G — {u1, U2, in, x12} — x13x14 and T' = T — {a0, a1, ar-1, ar}. Then e(G') > e(G) — 4(k — 1) + 1 > 1 (k2 — 6k + 2), which implies avedeg(G') > (k2 — 6k + 2)/k > k — 6 and | V(T') |< k — 4. By the induction hypothesis, T' C G'. If /'(a2) = x13, /'(ar-2) = x14, let /(a1) = x12, /(a0) = xn, /(a^) = U1, /(ar-1) = L.-T. Yuan andX.-D. Zhang: On the Erdos-Sos Conjecture for graphs on n = k + 4 vertices 59 u2. Since mi hits all the neighbours of f '(ar-2) in G', f is T-extensible. For the rest cases, similarly, it is easy to find an embedding from T to G. (a.2). There exists a vertex, say u2, in {u2, m3, m4} such that it hits at least two vertices of {in, X12, X13, X14}, say m2 hits in and X13, or m2 hits in and xi2. If «2 hits in and X13, let G' = G — {«1,^2} — X11X12 — X12X13 — X13X14 and T' = T — {a0, a1}. Then e(G') > e(G) — 2(k — 1) + 1 — 3 > 2(k2 — 2k — 8), which implies avedeg(G') > k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. Hence if f'(a2) = X11 or X13, let f (a1) = m2; if f'(a2) = X12, let f (a2) = u1 and f (a1) = «2; if f'(02) = X14 and X13 G f'(V(T')), let f (04) = X13 and f (ao) = «2; if f'(02) = X14 and X13 G f'(V(T')), let f (v) = «1,f (01) = X13,f (ao) = «2, because there is a vertex v, f '(v) = X13 and u1 hits all the neighbours of f'(v) in G'. Therefore f is T—extensible. If «2 hits X11 and X12, let G' = G — {«1,^2} — X12X13 — X13X14 — X11X14 and T' = T — {a0,a1}. Then e(G') > e(G) — 2(k — 1) + 1 — 3 > 2(k2 — 2k — 8), which implies avedeg(G') > k — 4 and | V(T') |< k — 2. By the induction hypothesis, T' C G'. Hence if f'(a2) = X11 or X12, let f (a1) = m2; if f'(a2) = X13 or X14, let f (a2) = u1, f (a1) = m2, because u1 hits all the neighbours of f'(a2) in G'. Therefore f is T—extensible. (a.3). u hits exactly one vertex of {x11, x12, x13, x14} for i = 2,3,4. (i). There exist two vertices of {m2,m3,m4} such that they hit the same vertex in {i11, X12, i13, i14}, says both m2 and m3 hit i14. If m2 and m3 misses the same vertices, say, {i11, i12, i13, y}, then m2 hits u3. Further, if G[i11, i12, i13, y] contains at most three edges or has a vertex of degree 3, the assertion follows from Case 2.5.2.(A) or Case 2.5.2.(B.1). Therefore we can assume that y hits both X11 and X13. Let G' = G — {«2, «3, X11, X12} — i13y and T' = T — {ao, 01, ar-1, ar}. The assertion follows from Case 2.5.2. (B.2).(a.1). If m2 misses {i11, i12, i13, y1} and m3 misses {i11, i12, i13, y2} with y1 = y2, let G' = G — {«1, «2,«3,X14} — X11X12 — X12X13 and T' = T — {ao, 01, ar-1, ar}. Then e(G') > e(G) — 4(k —1) + 4 — 2 > 2 (k2 — 6k + 4), which implies avedeg(G') > k — 6 and | V(T') |< k — 4. By the induction hypothesis, T' C G'. Hence if f '(02) = X11 or X13, let f (01) = i14,f (ao) = «3 or «2 or let f (02) = «1,f (01) = «3 or «2. If f'(02) = X12, let f (02) = «1,f (01) = «3 or «2. If f'(02) = y1 or y2, let f (01) = «3 or «2. Since there is a choice which uses distinct vertices of {u1, u2, u3, i14} for any two vertices of {i11, i12, i13, y1, y2}, we can find an embedding from T to G. (For example, if f '(a2) = X11,f'(ar-2) = X13, let f (01) = X14,f (ao) = «3,f (0^-2) = «1,f (0^-1) = «2.) (ii). {m2,m3,m4} hits the different vertices of {i11, i12,i13, i14}. Without loss of generality, we assume that u2 hits i11 and u3 hits i13, m2 misses y1 and m3 misses y2. Let G' = G — {«1, «2,«3,X13} — X11X12 — X11X14 and T' = T — {ao, 01, ar-1, ar}. Then e(G') > e(G) — 4(k —1) + 3 + 0 — 2 > 2 (k2 — 6k + 2), which implies avede-(G') > k — 6 and | V(T') |< k — 4. By the induction hypothesis, T' C G'. Hence if f'(02) = X12 or i14, let f (a1) = i13 and f (ao) = m3, or let f (a2) = u1 and f (a1) = m2, if f '(a2) = y1 or y2, let f (a1) = u1, if f'(a2) = i11, let f (a1) = m2, Therefore f is T—extensible. For the rest cases, by the same argument, it is easy to find an embedding from T to G. (b). u1 hits one or two vertices of {m2,m3, m4}. Without loss of the generality, we assume that u1 hits m2 and u1 misses u4. Then m4 g {i11, i12, i13, i14}, say m4 = i14, m4 misses u1,x12,z1,z2. If «2 = Z1, Z2, then «2 hits «4. Let G' = G — {«1, «2, «4, X12} — Z1Z2 and T' = 60 Ars Math. Contemp. 13 (2017) 107-123 T - ja0, a1, ar_i, ar}. Then e(G') > e(G) - 4(k - 1) + 2 - 1 > ± (k2 - 6k), which implies avedeg(G') > k - 6 and | V(T') |< k - 4. By the induction hypothesis, T' C G'. Hence if f '(a2) = x11 and f '(ar_2) = x13, let f (a1) = «4,f(ar_2) = u1 and f(ar_1) = «2, if f'(a2) = Z1 and f'(ar_2) = Z2, let f(a1) = «1,/(ar_2) = «4 and f (ar_1) = u2. Therefore f is T-extensible. For the rest cases, it is easy to find an embedding from T to G. If u2 = z1 or z2, say u2 = z1, let G' = G - {u1, u2, u4, x12} and T' = T - {ao , a1, ar_1, ar }. This situation is much easier than the above case. (c). u1 misses all the vertices of {u2, u3, u4}. Without loss of generality, we assume «2 = xn, «3 = X12, «4 = X13. Let «2 miss {«1,X13, y1,y2}. If G[«1, X13,y1,^2] contains two, or three edges, or a vertex of degree 3, the assertion follows from Case 2.5.2 (A). and Case 2.5.2 (B.1). Hence we assume that «1 hits y1, y1 hits «4 = x13, «4 hits y2 and y2 hits «1. Hence the assertion follows from Case 2.5.2. (B.2). (a) and Case 2.5.2. (B.2).(b). (C). There exists a 1 < i < 4, say i = 1, such that G[x11, x12, x13, x14] contains five edges. Then we assume that x11 hits x12, x13 and x14. Let G' = G - {«1, x11} -E(G[x12, x13, x14]) and T' = T - {a0, a1}. The assertion follows from the proof of Case 2.5.2 (B.1). (D). There exists a 1 < i < 4, say i = 1, such that G[x11, x12, x13, x14] contains six edges. If dG(x11) < k - 2, similar as Case 2.5.2 (B.1), we can prove the assertion. 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ARS MATHEMATICA CONTEMPORANEA 13 (2017) 63-79 The JLS model with ARMA/GARCH errors Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia Matjaž Omladic Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia Received 16 October 2014, accepted 3 December 2014, published online 21 October 2016 Prior to crashes, the stock index price time series is characterised by the Log-Periodic Power Law (LPPL) equation of the Johansen-Ledoit-Sornette (JLS) model, which leads to a critical point indicating a change to a new market regime. In this paper, we describe the hierarchical diamond lattice, upon which the JLS model is derived, using the diamond lattice operation Di and derive the recursion for the coefficients of the growth function in a diamond lattice rooted at the main root vertex rm. Further, to verify the adequacy of the JLS model, we analyse the model's residuals and propose its generalization, using the ARMA/GARCH error model. We determine the ARMA/GARCH orders using the extended autocorrelation function (EACF) method and compare the results with those of the Akaike and Bayesian Information Criteria. Using the data for 33 major world stock indices we show, that proposed generalization of the JLS model in general performs better in predicting the market regime changes and has also the ability to recognise false bubble identification, indicated by the JLS model. Keywords: Graph operations, hierarchical diamond lattice, JLS model, financial bubbles and crashes, ARMA/GARCH errors. Math. Subj. Class.: 05C76, 05C10, 82B20, 62P20, 62M10, 91B84 *Xlab d.o.o., Pot za Brdom 100, SI-1000 Ljubljana, Slovenia E-mail addresses: spela.jezernik@gmail.com (Spela Jezernik Sirca), matjaz@omladic.net (Matjaž Omladic) Spela Jezernik Sirca Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 64 Ars Math. Contemp. 13 (2017) 107-123 1 Introduction Financial bubbles and crashes are fascinating events for academics and practitioners alike, and such occurrences are especially interesting in the field of econophysics. These events are extremely important because of their usually strong impacts on not only financial markets but also the global economy. Unfortunately, there is still no general agreement in the literature on what defines a financial bubble or crash. A financial bubble may be recognised as a long-term positive deviation of a financial asset's market price from its fundamental value. A crash may be defined as a sudden dramatic decline of market price over a short time period. Understanding the behaviour of financial markets and the relationship between financial bubbles and crashes may help to minimise the damage of the speculative bubbles that end up with crashes. Consequently, identifying a financial bubble and predicting its end has become very important issue in financial markets behaviour research. The Johansen-Ledoit-Sornette (JLS) model [14] was developed to describe the dynamics of financial bubbles and crashes. The model states that, prior to crashes, the mean function of a stock index price time series is characterised by a Log-Periodic Power Law (LPPL) equation that leads to a critical point indicating the change to a new market regime, either a large crash or a change in the average growth rate. The model assumes the presence of two types of agents in the market, namely a group of agents with rational expectations and a group of irrational agents with herding behaviours, and these agents potentially lead to the development of speculative bubbles. These agents are organised into a network in which each exists in only one of two possible states (buy or sell), while their trading actions depend on the opinions of other agents and on external influences. If the tendency of irrational agents to imitate their neighbours increases up to a certain critical point, then a large proportion of agents will be in the same state (sell) at the same time, thus causing a crash. In the JLS model, bubbles are characterised by faster-than-exponential price growth due to herding behaviours and imitation of irrational agents during the bubble period. The key parameter of the model is the critical time tc, which is interpreted as the moment at which the bubble ends and the transition to another market regime begins. Numerous empirical results have been reported by several authors on this subject. The JLS model has been used in various types of markets, such as the bubbles of stock market indices [14, 16, 9, 44, 12], the anti-bubbles in different financial markets [15, 32, 40], exchange rate bubbles [20], the oil bubble [31], real estate bubbles [42, 43], corporate bond spread bubbles [3], credit default swap (CDS) spread bubbles [36], and the repo market size [38]. Most of the published research papers on the JLS model have focused on the existence of log-periodic fluctuations by fitting LPPL equation to the data. Although some papers have included several statistical methods for the detection of log-periodicity [14,29,6, 39,41,2], only a few papers have focused on the JLS model residuals [8,19,12]. The aim of this paper is to propose the JLS model generalization, based on the analysis of the JLS model residuals. Specifically, we investigate the presence of ARMA/GARCH patterns in the JLS model residuals, wherein we also compare our results with the log-peri-odic-AR(1)-GARCH(1,1) specification, proposed in [8]. According to ARMA/GARCH model determination, we examine the adequacy of the JLS model. In doing so, we compare the critical time parameter estimates, calculated with the JLS model, versus those calculated with generalized JLS model. We explore whether the generalized JLS model improves the JLS model estimates of the critical time parameters. To assure the generality of the results, we perform an analysis on large number of data samples. The rest of the paper is organised as follows. Section 2 describes derivation of the S. J. Sirca and M. Omladic: The JLS model with ARMA/GARCH errors 65 JLS model, together with the q-state Potts model and growth functions on hierarchical diamond lattice. In this section we also present details of the JLS model estimation, optimisation and verification. Our proposed generalization of the JLS model, methodology for the ARMA/GARCH model determination and parameter estimation are described in section 3. The data, empirical results of our analysis and main contributions of this paper are reported in section 4. Section 5 concludes the paper. 2 The Johansen-Ledoit-Sornette model 2.1 Motivation Financial markets consist of numerous interacting traders, that differ in size from small individuals to large institutional traders, such as pension funds. Moreover, all traders worldwide are organised inside a social network (family, friends, etc), within which they locally influence each other. The structure of financial markets resembles to hierarchical systems with traders on all different levels of the market. To develop the Johansen-Ledoit-Sornette (JLS) model, a model of rational imitation, Johansen et al. [14] used hierarchical diamond lattice representation for the structure of financial markets. In the case of hierarchical diamond lattice discussed by Berker and Ostlund [1], the lattice is generated in an iterative manner as shown in Figure 1. This is quite realistic model of complicated network of interactions between traders. The model is derived using the q-state Potts model on hierarchical diamond lattice defined in [5], where free energy exhibits log-periodic oscillations as the critical point is approached. For more details, see [13]. IO Figure 1: First few steps of building a hierarchical diamond lattice. 2.2 The q-state Potts model on hierarchical diamond lattice Let G be a graph and consider a set {1,2,..., q} of q elements, called spins. A state of a graph G is an assignment of a single spin to each vertex of the graph. Denote by V (G) = {v\,..., vn} vertex set of G and by E (G) edge set of G. Then the state of G is a function a : V (G) ^ {1,2,..., q}. Let S (G) denote the set of states of G. The interaction energy may be thought of simply as weights on the edges of the graph G. Denote by Je = J^j = JVi,Vj interaction energy on an edge e = {vj, Vj}. Then the Hamiltonian is h (a) = - Jj,jS (aj, aj), {i,j}eE(G) where a is a state of graph G, ai is the spin at vertex vi, S is Kronecker delta function and 66 Ars Math. Contemp. 13 (2017) 107-123 each edge {i, j} is assigned an interaction energy Jjj. Let the spins be positioned on a hierarchical diamond lattice constructed by the iterative process shown in Figure 1. Denote by Gn graph on the nth level of hierarchical diamond lattice. Then the q-state Potts model partition function on the nth level is defined as Zn (Gn)= £ e-^. aeS(Gn) The Potts model partition function is the sum of all possible states of an exponential function of the Hamiltonian. There exists a connection between the Potts model, which is useful to study phase transitions and critical phenomena in physics, and the graph theory, for example the Tutte polynomial [35] or Chromatic polynomial [28]. The graph theory is mathematical area, useful to describe and study the relations between participants in networks, such as physical, biological, social or economic networks. For some basic concepts used in models of economic networks, see [17]. Figure 1 depicts the first few steps of the diamond lattice operation, denoted by Di, where the original graph is a single edge. The graphs in the Figure are G0 = K2, Gi = Di(G0), G2 = Di(Gi), G3 = Di(G2). The original graph is planar, so is each next graph in the sequence. Di can be applied on any map (graph embedded on a surface). Note that Di is a composite operation. In [7, 24] several operations on maps are considered. One, Pa, parallelization replaces each edge by a pair of parallel edges and another one Su1, one-dimensional subdivision subdivides each edge of the original map. In this way Di(M) = Su1(Pa(M)). Note that Di may be regarded as an operation on map or due to its simplicity also as an operation on the underlying graph. A theory of representations of graphs and maps has been laid down by Pisanski and Zitnik [26], where such operations were considered. Repeated operations were used in other contexts, see for instance [25]. Operations on maps have been studied in connection with symmetry in several papers, see for instance [23, 22, 4, 10]. 2.3 Growth function in rooted diamond lattice Recall the definition of growth function in rooted graphs [25]. Let G be a connected, finite or locally finite (infinite graph with finite vertex degrees) graph. Let V(G) be the set of vertices of G and let r e V(G) be the so-called root of graph G. The distance d(u,v) between u, v e V(G) is defined as number of edges on the shortest path between u and v. Further, we define the (spherical) growth sequence as {¿(G, r, n)|n = 0,1,2,... }, where S(G, r, n) denote the number of vertices at distance n from root r in graph G. Then the (spherical) growth function of graph G rooted at r can be written as follows: f (G,r,x) = £ ¿(G,r,n)xn, n=0 i.e. the generating function for S(G, r, n) of graph G at root r. Here we limit to the case when the root is single vertex, vertex root, although we can extend the definition by allowing that a root is any induced subgraph of graph G. We start the construction of hierarchical diamond lattice with a graph G0 = K2. Its growth function is f (G0, r, x) = 1 + x and is independent of the selected root r. Using the diamond lattice operation on graph G0 we get graph Gi = Di(G0), for which S. J. Sirca and M. Omladic: The JLS model with ARMA/GARCH errors 67 f (Gi, r, x) = 1 + 2x + x2 is also independent of r. If we use diamond lattice operation at least two times, we have to calculate each of the growth functions separately depending on the root. Here we first limit ourselves to the case of selecting one of the two vertices at the top and bottom in each graph Gn, called main root vertex, denoted by rm (white vertices of graph Gn on Figure 1). Note that since each diamond lattice Gn can be mirrored horizontally (vertically for n > 0), horizontally (vertically) symmetrically selected roots produce an identical growth function. In the next subsection we examine the cases of selecting vertex root different from rm in Gn for n = 1,2, 3,4. The results for all possible growth functions of a graph Gn for n > 4 will be published elsewhere. Let us present the growth function of graph Gn rooted at rm for n = 2 and n = 3: f (G2, rm, x) = 1 + 4x + 2x2 + 4x3 + x4 f (G3,rm,x) = 1 + 8x + 4x2 + 8x3 + 2x4 + 8x5 + 4x6 + 8x7 + x8. If we denote the growth function of graph Gn rooted at rm by gn(x), we can write the system of equations that determine the growth functions f (Gn, rm, x) as follows: go(x) = (1 + x) gi(x) = (1+ x)(2go(x) - (1 + x)) g2(x) = (1+ x2)(2gi(x) - (1 + x2)) 53(x) = (1+ x4)(2g2(x) - (1 + x4)) gn(x) = (1+ X2" )(2gn_i(x) - (1 + : (2.1) Moreover, if we define the function /o(x) = 1 and /n(x) = ]^[(1 + x2") for n > 0, i=i (2.2) we can write the recursion of the growth function /(Gn, rm, x) also as: gn(x) = gn_l(x2) + 2"x/„_i(x). To derive the recursion for the list of coefficients of gn (i.e. for the growth sequence of graph Gn rooted at rm), we first define some operations on lists. Let w = {wi, w2,..., wm} and w = {wW1, w2,..., wn} be two lists of coefficients. Let a(k) = {a, a,..., a} denote the list of k repetitions of the value a. Then we define the following operations on the lists: t(w, w m(w, k v(w, a b(w c(w = w + w = {wi, w2, . . . , wm, wi, w2, . . . , wn} = w • k = {wik, w2k,. .., wmk} =wa (m_i) = = { wi , a, w2 , ,wm_ i, a, w„ w[ 1] = {w2, w3, . . . , wm} w[ 1, m] = {w2, w3, .. ., wm_i} . Let an denote the list of coefficients of gn. Then we derive the following rule for generating the list of coefficients for gn from the list of coefficients for gn_i and g0: ao = {1, 1} an = t(6(ao),t(m(t(6(a„_i),c(a„_i)), 2),b(ao))). (2.3) 2 } 68 Ars Math. Contemp. 13 (2017) 107-123 Alternatively, the recursion in (2.3) can be written as follows: a„ = v(a„_i, 2n). (2.4) The first four lists an of coefficients for gn are: ao = {1, 1} ai = {1,2,1} a2 = {1, 4, 2,4,1} a3 = {1, 8,4, 8, 2, 8, 4, 8,1} At each step we get a list an of coefficients for gn of length 2n + 1. 2.4 Different growth functions of Gn for n = 1, 2, 3, 4 In the previous subsection we examined the growth function in diamond lattice Gn rooted at rm. Here we examine the number of different growth functions in Gn and calculate the growth functions separately depending on the root r different from rm in Gn for n = 1,2,3,4 (the graph G0 has only two main root vertices). For a graph G and v e V(G), the degree of a vertex v is denoted by dG(v) and the degree set of a graph G (i.e. the set of (distinct) degrees of the vertices of G) by D(G), where we write the degree set in ascending order. Then, we define the set VD(G) = [pki(G)|kj e D(G), i = 1, 2,..., |D(G)|}, where pk(G) denotes the number of vertices of a degree k in G. Let A be the automorphism group of G that partitions the vertex set V(G) into orbits: V(G) = [vi] U [v2] U • • • U [vs], where the number of orbits of G is |Orb(G) | = s. If u and v belong to the same orbit of Gn, then f (Gn, u, x) = f (Gn, v, x). We provide in Table 1 a number of different growth functions in Gn for n = 1,2,3,4. Additionally, we provide the number of vertices and edges, degree set and set of number of vertices with the same degree. n |V (Gn) | |E(G„)| D(Gn) VD(Gn) |Orb(Gn)| 1 4 4 {2} {4} 1 2 12 16 {2, 4} {8,4} 2 3 44 64 {2,4, 8} {32, 8,4} 3 4 172 256 {2,4, 8,16} {128, 32, 8,4} 5 Table 1: Number of vertices and edges, ascending ordered degree set, set of number of vertices with the same degree and number of orbits in diamond lattice Gn for n = 1, 2, 3,4. It is not hard to see that |V (Gn)| |E(Gn)| D(Gn) VD(Gn) 2 = 3(2 + 4n) = 4n = {2^ = 1, 2,...,n} (n-i)+1 |i = 1, 2, , n — 1 } U {22} . S. J. Sirca and M. Omladic: The JLS model with ARMA/GARCH errors 69 If we are given a root r in Gn this defines a root r in the corresponding Dik (Gn). It is not hard to see that, if dGn (r) = 2® for i e {1, 2,..., n}, then dDi(Gn)(r) = dGn+1 (r) = 2i+1. We now consider the cases of all possible growth functions of a graph Gn (1 < n < 4) when the root r is a single vertex. First we examine root r of graph Gn which produces an identical growth function as main root vertex rm. If r e V(Gn), r = rm and dGn(r) = dGn (rm),then f (Gn,r,x) = gn(x) and(rm)(Gn) = 4. That means only four vertices in graph Gn (n > 1) produce an identical growth function gn(x) (black and white vertices on Figure 1). Next, for n = 2,3,4 the growth functions of a rooted graph Gn at r, where dG2 (r) = 2, dG3 (r) = 4, dG4 (r) = 8 (red vertices on Figure 1) are: f (G2,r,x) = 1 + 2x + 5x2 + 2x3 + 2x4 f (G3, r, x) = 1 + 4x + 2x2 + 12x3 + 5x4 + 8x5 + 2x6 + 8x7 + 2x8 f (G4, r, x) = 1 + 8x + 4x2 + 8x3 + 2x4 + 24x5 + 12x6 + 24x7 + 5x8 + 16x9 +8x10 + 16x11 + 2x12 + 16x13 + 8x14 + 16x15 + 2x16. Moreover, using the polynomial p(x) = 1 + 3x + 2x2 + 2x3 and function fn (x) defined in (2.2), we can write the following system of equations that determine the growth functions of a rooted graph Gn at r, where dGn (r) = 2n-1: f (G3,r,x) = f (G2,r,x2)+22xfo(x)p(x2) f (G4,r,x) = f (G3,r,x2)+23xf1(x)p(x4) f (Gn, r, x) = f (Gn-1,r,x2) + 2n-1xfn-3(x)p(x2n-2). For n = 3,4, the growth functions of a rooted graph Gn at r, where dG3 (r) = 2 and dG4 (r) = 4 (green vertices on Figure 1) are: f (G3, r, x) = 1 + 2x + 9x2 +4x3 + 10x4 + 3x5 + 8x6 + 3x7 + 4x8 f (G4, r, x) = 1 +4x + 2x2 + 20x3 + 9x4 + 16x5 + 4x6 + 24x7 + 10x8 + 16x9 +3x10 + 16x11 + 8x12 + 16x13 + 3x14 + 16x15 + 4x16, = f (G3, r, x2)+4xq(x2), where q(x) = 1 + 5x + 4x2 + 6x3 + 4x4 + 4x5 + 4x6 + 4x7. Finally, we examine the root r of graph G4 with dG4 (r) = 2, which belongs to one of two orbits. We denote the root in the first orbit by r1 (ur1 e E(G4) and dG4 (u) = 16) and in the second orbit by r2 (ur2 e E(G4) and dG4 (u) = 8). Then the growth functions f (Gn, r®, x) for n = 4 and i = 1,2 are: f (G4, r1, x) = 1 + 2x +17x2 +8x3 + 18x4 + 5x5 + 16x6 + 7x7 + 20x8 + 5x9 +16x10 + 6x11 + 16x12 + 6x13 + 16x14 + 5x15 + 8x16 f (G4, r2, x) = 1 + 2x + 9x2 +4x3 + 16x4 + 7x5 + 24x6 + 9x7 + 20x8 + 6x9 +16x10 + 5x11 + 16x12 + 5x13 + 16x14 + 8x15 + 8x16. 2.5 Derivation of the JLS model The structure of financial markets, given by hierarchical diamond lattice, can be described as follows. Start with a two linked traders. Secondly, replace this link by a diamond, where 70 Ars Math. Contemp. 13 (2017) 107-123 the original traders occupy two diametrically opposite vertices, and the two other vertices are occupied by two new traders. Thirdly, for each of these four links, replace it by a diamond in the same way. If we repeat this process, we get a hierarchical diamond lattice and after n iterations we have § (2+ 4n) traders and 4n links among them. Johansen et al. [14] constructed so-called JLS model, a non-linear model suitable for the prediction of crash time in both microscopic and macroscopic modelling. This model characterises the occurrence of a crash by the crash hazard rate h (t). All traders are organised into a network and they locally influence each other through this model. The model assumes that agents tend to imitate the opinions of their nearest neighbours. The imitation process is described by the crash hazard rate h (t) with a power law, i.e. dh/dt = Chs, where C is a positive constant and S > 1. In the JLS model [14] we consider a network of traders, where each trader is indexed by an integer number i = 1,... ,N and N (i) denotes the set of traders who are directly connected to trader i in the network (hierarchical diamond lattice). For simplicity we consider a special case of the Potts model for q = 2. We assume that trader i can be in only one of two possible states at time t: the buy (si = +1) or the sell (si = -1) decision. Then the state of trader i is determined by the following Markov process: si = sign ( K sj + aei + G ^ jew(i) where the sign function, sign (x), is equal to +1 (-1) if x > 0 (x < 0), K is positive constant, ei ~ N (0,1) is an independent and identically distributed random variable and term G is a measure of some external influence, which tends to favor state +1 (-1) if G > 0 (G < 0). In this model, K governs the tendency of imitation among traders, while a governs their idiosyncratic behaviour. If we define the average state of the market as M = (1/I)J2[=i si, the susceptibility of the system is defined as x = dEdcM] and measures the sensitivity of the average state of the system to a small external influence. Further the JLS model assumes that the crash hazard rate behaves in a similar way as the susceptibility in the neighbourhood of a critical point. By considering a hierarchical diamond lattice for the financial market, the dynamics of the crash hazard rate can be described as follows: h (t) « B0 (tc - t)m-1 + C0 (tc - t)m-1 cos [w ln (tc - t) - ^], where B0, C0 and w are real number. The dynamics of the price is described as dp = ^ (t) p (t) dt - Kp (t) dj, where p (t) is the price, ^ (t) is time-varying drift and j is a jump process, such that dj = 0 before crash and dj = 1 after the crash occurs at critical time tc. The parameter k determines the loss amplitude associated with the occurrence of a crash. One assumption of this model is that the price p(t) follows a martingale process, i.e. Et [p(t )] = p (t), Vt > t, where Et [•] represents the conditional expectation on all information available up to time t. Then we have Et [dp] = 0. The dynamics of the jumps is governed by a crash hazard rate h (t) and Et [dj] = h (t) dt. Furthermore, the drift ^ (t) is chosen so that the martingale condition is satisfied, which yields ^ (t) = Kh (t). Then the simplest form of a log-price dynamics up to the end of the financial bubble can be written as follows: ln(pt ) = A + B (tc - t)m + C (tc - t)m cos (w ln (tc - t) - 0) + ut, (2.5) S. J. Sirca and M. Omladic: The JLS model with ARMA/GARCH errors 71 where pt is the price of a stock index or some other specific asset at time t, A = ln (ptc) > 0 is the logarithm of the price at the critical time tc and B < 0 for a growing bubble before the crash. The critical time tc is the end of a bubble and indicates a change to a new market regime, which could be a large crash or a change in the price growth rate. Note that tc is the most probable time for the crash, but there also exists a nonzero probability that the bubble ends without a crash. If C = 0, then the presence of log-periodic behaviour is indicated. The exponent m lies between 0 and 1 to ensure a finite price, even at tc. Parameter w is the frequency of oscillations during the bubble period, while ^ is a phase parameter and lies between 0 and 2n. The error term ut has a zero mean and some standard deviation. 2.6 Estimation (optimisation) of the JLS model The JLS model in equation (2.5) is described by three linear parameters, A, B and C, and four nonlinear parameters, tc, m, w and For simplicity, we denote the 7-dimensional vector of these parameters by 0 = [A, B, tc, m, C, w, We need to find the vector 0 that best fits the observed log-price time series {ln (p^}^ within the estimation time period [ti, tn], where tn < tc. Although different measures can be used, the most common approach is to minimise the sum of the squared residuals: where ut is the error term in the JLS model. The minimisation of such an objective function S (0) is not an easy task due to the presence of many local minima. Therefore, we first reduce the number of parameters by expressing three linear parameters as a function of the remaining four nonlinear parameters, as proposed in [14]. If we rewrite the equation (2.5) as ln (pt) « A + Bft + Cgt, where ft = (tc - t)m and gt = (tc - t)m cos (w ln (tc - t) - then we obtain the estimates of linear parameters A, B and C by using the ordinary least squares (OLS) method: If we rewrite the system of equations (2.7) in matrix form as XTy = (XTX)^, where then the well-known solution is / = (XTX)-1XTy. By reducing the number of parameters from seven to four we decreased the complexity of the optimisation problem. However, we still need to find the global minimum in a 4-dimensional space of the objective function: (2.6) (2.7) Si(0i) = min S(0), A,B,C (2.8) where 01 = [tc, m, w, denotes the vector of nonlinear parameters and S(0) is given in equation (2.6). 72 Ars Math. Contemp. 13 (2017) 107-123 Many different optimisation algorithms have been proposed to estimate the JLS model [30, 14, 11, 27, 8, 18]. In this paper, we use the Differential Evolution (DE) algorithm, which is a simple and efficient population-based search heuristic developed by Storn and Price [33]. We employ the DEoptim function in the R package DEoptim (see [21] for more details) together with the following intervals for each parameter's optimisation: where t\ and tn are first and last data point in the estimation time period. These intervals are similar to those used in [37]. If the parameter m is too small, then we obtain a bubble with a sudden acceleration at the end, while too large an m corresponds to an almost linear non-accelerating bubble. Similarly, if the frequency w of the oscillations is too small, then the log-periodic oscillations will be too slow, while for too large a value of w, the oscillations are too fast. Therefore, after the estimation of the JLS model, we accept all results that satisfy the following four additional constraints: B < 0, 0.1 < m < 0.8, 4 < w < 15 and tc < 2tn - tx. (2.11) Similar bounds for m and w were used in [29]. It is also reasonable to eliminate the results, for which the upper bound of the search space for tc under (2.9) is attained. Therefore we use the last constraint under (2.11). For the sake of brevity, we refer to conditions (2.11) as the LPPL conditions. 2.7 Verification of the JLS model Important step in building a model is determination of its quality. If the model specification is appropriate, then the residuals should behave like true stochastic components. If this component is white noise, then the residuals should behave like independent (normal) random variables, with a zero mean and some standard deviation. To investigate the stationarity of the residuals ut in the JLS model (2.5), we perform Phillips-Perron (PP) and Augmented Dickey-Fuller (ADF) unit root tests, where the null hypothesis is the presence of a unit root. Additionally, the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test is employed for testing the null hypothesis, which posits that the observable time series is stationary. We perform the Shapiro-Wilk test on the residuals as a formal test of normality. We are also interested in searching for possible dependencies in the JLS model residuals. We apply a runs test to verify the independence of the residuals. Using the autocorrelation function (ACF) of the residuals and squared residuals, we can observe the presence of linear and nonlinear dependence in the residuals. However, for a mixed autoregressive moving average (ARMA) model, it is usually difficult to identify the ARMA orders p and q from these plots. Tsay and Tiao [34] proposed an extended autocorrelation function (EACF) method for model identification that is able to identify mixed ARMA(p, q) models, as well as pure AR(p) and MA(q) models. Details about the EACF method and our proposed algorithm for ARMA order determination can be found in supplementary file (section 1). We first apply the EACF method on the JLS model residuals to select the appropriate ARMA model. Then we compare this result with two of the information criteria, tc G [tn + 1, 2tn - ti + 1] , m G [10-5,1 - 10-5] w G [10-5, 40] , ^ G [10-5, 2n - 10-5] (2.9) (2.10) S. J. Sirca and M. Omladic: The JLS model with ARMA/GARCH errors 73 namely Akaike's Information Criterion (AIC), given by AIC = log (a2) + 2k, and Schwarz Bayesian Information Criterion (BIC), given by BIC = log (a2 ) + n log (n), where k is the number of estimated parameters, a2 is the estimated error variance of the fitted model, and n is the number of observations. We estimate the parameters of all ARMA(p, q) models with 0 < p,q < 5 and select the ARMA orders by minimising AIC and BIC, respectively. For this purpose, we employ an arima function in the R package TSA. 3 The JLS model with ARMA/GARCH errors Gazola et al. [8] proposed a log-periodic-AR(1)-GARCH(1,1) model to capture the structure of the JLS model residuals. According to the analysis of residuals described in subsection 2.7, our empirical results on different stock indices show similar results, namely, that the residuals are strongly autocorrelated and in many cases also heteroscedastic. Furthermore, by using the EACF method we also found that an ARMA model is sometimes more appropriate than a pure AR model. Consequently, to capture the behaviour of the error term ut in (2.5), we propose the following ARMA(p, q)/GARCH(P, Q) error model: p q ut = E Pi Ut-i + nt + E^j nt-j, (3.1) i=i j=i nt = atet, (3.2) p Q o\ = ao + J2 aknt-k + E ^lat-l, (33) k=1 1=1 where et is independent and identically distributed process with a zero mean and unit variance. If q = 0 in equation (3.1), we have a pure AR(p) process and if p = 0 we have a pure MA(q) process. We also include the conditional variance a2, which evolves according to a GARCH(P, Q) process described by equation (3.3). If P = Q =0 in equation (3.3), we have a constant variance and obtain a pure ARMA(p, q) error model. 3.1 Estimation of the JLS model with ARMA/GARCH errors Let us denote by 0 = [A, B, tc, m, C, w, pi, Qj] the vector of the parameters of the JLS model with ARMA(p, q) errors and by 01 = [A, B, tc, m, C, w, pi, Qj,ak, $], the vector of the parameters of the JLS model with ARMA(p, q)/GARCH(P, Q) errors, where i = 1,..., p, j = 1,..., q, k = 0,..., P, and l = 1,..., Q. In our empirical analysis, we use the following three-step procedure for model identification and estimation of the JLS model with ARMA/GARCH errors: 1. Estimate the JLS model (2.5), as described in subsection 2.6 and verify that the LPPL conditions under (2.11) are satisfied. 2. Identify the ARMA orders with EACF method (compare with AIC and BIC) as described in subsection 2.7 and estimate the JLS model with ARMA(p, q) errors. Using the conditional maximum likelihood method, under the normality assumption for et in equation (3.2), the estimates of parameters 0 are obtained by the maximization of the log-likelihood function: L (0) = -^ log(2n) - ^ log(a2) - £ , (3.4) t=p+i 74 Ars Math. Contemp. 13 (2017) 107-123 where rjt and a2 = a2 are obtained with equation (3.2), and n is the length of the residuals series. Perform the Engle's Lagrange Multiplier (LM) test (for lags 10, 12 and 20) to verify the presence of autoregressive conditional heteroscedasticity (i.e. ARCH effect) in the residuals from the estimated model in the previous step. If necessary, identify the GARCH orders with EACF method (compare with AIC and BIC) on the squared and absolute residuals. The maximum likelihood estimation for the JLS model with ARMA(p, q)/GARCH(P, Q) errors can be carried out by maximizing the log-likelihood function: 1 n 1 n 2 L (0i) = -n-p log(2n) - 2 £ log(o?) - 2 £ , (3.5) t> o a2 --2 - U + t = p+1 t = p+1 t where n2 and are obtained with equation (3.2), and n is the length of the residuals series. For simplicity, if there is an ARCH effect in the residuals, we incorporate a simple GARCH(1,1) model. That means, we are not interested in particular GARCH order, but only in the existence of GARCH structure in the residuals. Therefore, we propose new algorithm for determination if there exist an ARCH effect; see section 1 in supplementary file. Some additional comments on described three-step procedure can be found in supplementary file (section 2). 3.2 Verification of the JLS model with ARMA/GARCH errors If our proposed model specification is appropriate, than the standardized residuals et = nt/Ut are approximately independent and identically distributed. Therefore, we proceed with the analysis of the standardized residuals from the fitted JLS model with ARMA errors (the model estimated in step 2 of three-step procedure described in previous subsection), if the results in step 3 show that there is no need to incorporate a GARCH process. Otherwise we perform the analysis of the standardized residuals from the fitted JLS model with ARMA/GARCH errors. The standardized residuals are investigated using the same set of tests as described in subsection 2.7. 4 Empirical results 4.1 The data The dataset consists of daily closing prices for 33 major stock indices worldwide, namely 10 American, 12 European and 11 Asian/Pacific indices. Detailed information about the dataset can be found in supplementary file (section 3). The data were downloaded from Yahoo! finance to the 28th July 2014. According to the available data, we selected stock indices that represent trading activities on main stock exchanges in each geographic region to cover. We did so to cover adequate volume of trading activity in order to assure the generality of our results. 4.2 The JLS model estimation We apply the procedure as described in section 3 to estimate the JLS model parameters. For all stock indices in the dataset, we generate a set of time windows, each consisting of S. J. Sirca and M. Omladic: The JLS model with ARMA/GARCH errors 75 500 successive trading days (which is approximately two calendar years). Note, the JLS model is usually estimated on 2-3 years large time window. Each set is obtained by rolling such time windows over the whole dataset with increment of 1 day. To summarize, the results of the JLS model application to rolling time windows for which the LPPL conditions, as specified in (2.11), are satisfied show, that fraction of such time windows varies between 15.14% and 2.39%. We note that such variation among stock indices reflects also different starting points of the dataset and therefore different bubbles representation across stock indices. To increase the possibility that the selected samples (for further analysis) resemble the bubble periods, we set the minimum number of successive time windows (that satisfy LPPL conditions) to 50. For more details, see section 4 in supplementary file. 4.3 The JLS model residual analysis In this subsection we investigate the JLS model residuals ut in (2.5) and estimate the JLS model with ARMA/GARCH errors. We perform analysis of 30 stock indices (121 selected samples). For three indices no sample is selected. Detailed information about the selected samples can be found in supplementary file (section 5). To summarize, results shows that PP, ADF and KPSS tests indicate the stationarity for most cases at the 5% level. For each stock index, we examine any possible dependence in the JLS model residuals by performing a runs test on all time windows of selected samples. For all cases we get p-values of < 0.05, therefore we can reject the null hypothesis of independence at the 5% level. The Shapiro-Wilk test rejects the normality assumption in most cases at the 5% level. For more details, see section 6 in supplementary file. To see how Gazola et al. [8] proposed model specification holds in the case of our selected data, we first investigate the presence of ARMA orders, different from ARMA(1,0) in selected samples. According to the EACF method the results indicate that the fraction of ARMA(1,0) models varies significantly across the stock indices. We found that, contrary to findings in [8], for only four stock indices the EACF method suggest only the ARMA(1,0) model, while for two indices it suggests only ARMA models with p, q > 0. In most cases the BIC method confirms the results of EACF method, while in general the selected ARMA orders with AIC method are greater compared to those selected by BIC and EACF methods. For more details, see section 7 in supplementary file. Next, we investigate the existence of GARCH structure in the residuals. We use the following strategy to decide, whether there is no need to incorporate GARCH process: if at least one EACF result on squared or absolute residuals suggest that P = Q = 0 and LM test for at least two lags (10, 12 or 20) confirms that results, or LM test results cannot reject the null hypothesis at the 5% level for all three lags, then the JLS model with only ARMA errors is selected. The results show that, contrary to findings in [8], only for about third of all analysed stock indices the incorporation of GARCH is always necessary and also that the biggest proportion of such indices is in Asian/Pacific region. Here we note that the results comparison between different geographic regions is limited due to different historical data availability. For more details, see section 7 in supplementary file. As a last step, we proceed with the analysis of the standardized residuals from our proposed generalized JLS (GJLS) model. The Shapiro-Wilk test on standardized residuals of the GJLS model shows quite similar result as JLS model residual analysis. There exists only one index for which the fraction of rejected normality assumption is smaller than 85%. Similarly, in most cases, the PP, ADF and KPSS indicate the stationarity of the 76 Ars Math. Contemp. 13 (2017) 107-123 standardized residuals at the 5% level. The key difference between analysis of the JLS and GJLS model residuals is obtained using a runs test. For 16 indices, performing this test yields fraction smaller than 5% of the rejected null hypothesis at the 5% level. Note that for almost all indices, where this fraction is larger than 5%, using the GJLS model we identify the false JLS model bubble identification. Detailed explanation of the false JLS model bubble identification can be found in supplementary file (section 9). 4.4 The JLS model versus GJLS model In this subsection we compare the results of the JLS and GJLS model. In doing so, we focus on the parameter of our key interest, critical time tc. For all 121 selected samples, we compute two location parameters for the distribution of tc, namely the mean and the median, and 25-75% quantile interval for tc. For some explanatory comments on the estimation methodology; see section 8 in supplementary file. Our main results show that GJLS model in general performs better in predicting the actual local or global market peak, denoted by tc obs, which is consistent with findings in [8], that the GJLS model outperforms the JLS model in predicting tc. For the American stock indices the mean or median estimate of the GJLS model is 26 times closer to tc obs, while the JLS model only 14 times. Similar results are also obtained for the European and Asian/Pacific stock indices; see section 9 in supplementary file. Estimated 25-75% quantile intervals (QI) for tc (calculated with the JLS or GJLS model) cover the tc obs for more than half (sub)samples for stock indices in all three geographic regions. The results differ, however, when comparing the JLS and GJLS models; in the case of the American stock indices the coverage for the JLS model is 14 times, while for the GJLS 18 times. There are also 6 cases for each model, where the estimated QI misses the tc obs for less than 11 trading days. For the European stock indices the QI coverage is 17 (25) times for the JLS (GJLS) model and the number of estimates outside the interval by less or equal to 2 weeks is 11 (12). For the Asian/Pacific stock indices the JLS (GJLS) model QI covers the tc obs 5 (10) times, with 3 estimates outside the intervals for less or equal to 2 weeks for each model. For more details, see section 9 in supplementary file. Comparing the mean and median estimates across the JLS and GJLS models for the American stock indices, the former performs better in 24 cases, while the later in 14 cases. Similar outcome is observed also for the European stock indices. For Asian/Pacific indices the mean and median perform almost equally good, but we note that for this region we have considerably lower number of (sub)samples. It therefore makes sense to take into account also the choice of the point estimate measure. For more details, see section 9 in supplementary file. 5 Conclusions In this paper we consider the Johansen-Ledoit-Sornette (JLS) model, which describes the behaviour of stock index prices during a bubble regime and also predicts the most probable time for a change in the regime. We introduce the diamond lattice operation Di which is a composite operation of parallelization and one-dimensional subdivision, to describe the construction of the hierarchical diamond lattice used in JLS model derivation. It turns out that the operation Di has interesting properties that we are investigating. The results will be published elsewhere. The idea of our JLS model generalization is motivated by the behaviour of the JLS S. J. Sirca and M. Omladic: The JLS model with ARMA/GARCH errors 77 model residuals. The results of our analysis reveal the presence of a strong autocorrelation in JLS model residuals and heteroscedasticity in many cases. To incorporate these two properties into the model, we propose an ARMA(p, q)/GARCH(P, Q) error model and a methodology for model identification, parameter estimation and its verification. As the first part of our analysis we investigate the behaviour of the JLS model residuals. To assure the generality of the results, we perform an analysis over a large size of samples for 33 stock indices from three broader world geographic regions. Our results suggest that there is no general rule for determination of ARMA/GARCH specification of the JLS model as the proportion of ARMA models, excluding ARMA(1,0) and proportions of ARMA models without GARCH specification varies significantly across the samples and also within each geographic region. We take this result as a reminder for careful ARMA/GARCH order determination, when analysing particular time period for stock indices. In the second part of the analysis we show that specified JLS model generalization outperforms the JLS model estimates of critical time tc. Using the mean value and the median of tc parameter estimates, the results show smaller deviations from the actual crash dates for the GJLS model. Moreover, we also show that 25-75% quantile intervals more often cover the tc parameter estimates of the GJLS model than for the JLS model. 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Wanless * School of Mathematical Sciences, Monash University, Vic 3800, Australia Received 9 June 2016, accepted 21 November 2016, published online 19 February 2017 Abstract An integer Heffter array H(m, n; s, t) is an m x n partially filled matrix with entries from the set {±1, ±2,..., ±ms} such that i) each row contains s filled cells and each column contains t filled cells, ii) every row and column sums to 0 (in Z), and iii) no two entries agree in absolute value. Heffter arrays are useful for embedding the complete graph K2ms+1 on an orientable surface in such a way that each edge lies between a face bounded by an s-cycle and a face bounded by a t-cycle. In 2015, Archdeacon, Dinitz, Donovan and Yazici constructed square (i.e. m = n) integer Heffter arrays for many congruence classes. In this paper we construct square integer Heffter arrays for all the cases not found in that paper, completely solving the existence problem for square integer Heffter arrays. Keywords: Heffter array, biembedding. Math. Subj. Class.: 05B20, 05C10 1 Introduction We begin with the general definition of Heffter arrays [1]. A Heffter array H(m, n; s, t) is an m x n matrix with nonzero entries from Z2ms+1 such that 1. each row contains s filled cells and each column contains t filled cells, 2. the elements in every row and column sum to 0 in Z2ms+1, and 3. for every x e Z2ms+1 \ {0}, either x or —x appears in the array. * Research supported by ARC grant #DP150100506. E-mail addresses: jeff.dinitz@uvm.edu (Jeffrey H. Dinitz), ian.wanless@monash.edu (Ian M. Wanless) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 82 Ars Math. Contemp. 13 (2017) 107-123 The notion of a Heffter array H(m, n; s, t) was first defined by Archdeacon in [1]. It is shown there that a Heffter array with a pair of special row and column orderings can be used to construct an embedding of the complete graph K2ms+1 on a surface. This connection is given in the following theorem. Theorem 1.1 ([1]). Given a Heffter array H(m,n; s,t) with compatible orderings ur of the symbols in the rows of the array and on the symbols in the columns of the array, then there exists an embedding of K2ms+1 on an orientable surface such that every edge is on a face of size s and a face of size t. Moreover, if wr and are both simple, then all faces are simple cycles. We refer the reader to [1] for the definition of a simple ordering and the definition of compatible orderings. We will not concern ourselves with the ordering problem in this paper and will concentrate on the construction of Heffter arrays. In [4] the ordering problem is addressed in more detail in the case when m = t = 3 and n = s. A Heffter array is called an integer Heffter array if Condition 2 in the definition of Heffter array above is strengthened so that the elements in every row and every column sum to zero in Z. In [2], Archdeacon et al. study the case where the Heffter array has no empty cells. They show that there is an integer H(m, n; n, m) if and only if m, n > 3 and mn = 0, 3 (mod 4) and in general that there is an H(m, n; n, m) for all m, n > 3. In this paper we will concentrate on constructing square integer Heffter arrays with empty cells. If the Heffter array is square, then m = n and necessarily s = t. So for the remainder of this paper define a square integer Heffter array H(n; k) to be an n x n partially filled array of nonzero integers satisfying the following: 1. each row and each column contains k filled cells, 2. the symbols in every row and every column sum to 0 in Z, and 3. for every element x G {1, 2,..., nk} either x or — x appears in the array. In [3] the authors study the case of square integer Heffter arrays H(n; k). The following theorem is from that paper. Theorem 1.2 ([3]). If an H (n; k) exists, then necessarily 3 ^ k ^ n and nk = 0, 3 (mod 4). Furthermore, this condition is sufficient except possibly when n = 0 or 3 (mod 4) and k = 1 (mod 4). It should be noted that [3] also contains partial results when n = 0 or 3 (mod 4) and k = 1 (mod 4). In this paper we will solve those cases completely. Our main result is given in the following theorem. Theorem 1.3. There exists an integer H (n; k) if and only if 3 ^ k ^ n and nk = 0, 3 (mod 4). We will prove this theorem by first constructing an H(n; 5) where all the filled cells are contained on 5 diagonals. Then we will add s disjoint H(n; 4) to construct H(n; 5 + 4s) = H(n; k) where k = 1 (mod 4). We begin in Section 2 by giving a general construction for H(n; 4) where all of the filled cells are contained in 4 diagonals. In Section 3 we discuss the case when n = 3 (mod 4) and in Section 4 we discuss the case when n = 0 (mod 4). J. H. Dinitz and I. M. Wanless: The existence of square integer Heffter arrays 83 2 H(n; 4) using two sets of consecutive diagonals An important concept in the prior work on Heffter arrays has been the notion of a shiftable Heffter array. A shiftable Heffter array Hs(n; k) is defined to be a Heffter array H(n; k) where every row and every column contain equal numbers of positive and negative entries. Let A be a shiftable array and x a nonnegative integer. If x is added to each positive element and —x is added to each negative element, then all of the row and column sums remain unchanged. Let A ± x denote the array where x is added to all the positive entries in A and —x is added to all the negative entries. If A is an integer array, define the support of A as the set containing the absolute value of the elements contained in A. So if A is shiftable with support S and x a nonnegative integer, then A ± x has the same row and column sums as A and has support S + x. In the case of a shiftable Heffter array Hs(n; k), the array Hs(n; k) ± x will have row and column sums equal to zero and support S = {1 + x, 2 + x,..., nk + x}. In this section we describe an easy construction of a shiftable H(n; 4) where all of the filled cells are contained in two pairs of adjacent diagonals. If H is an n x n array with rows and columns labeled 1,..., n, for i = 1,..., n define the i-th diagonal A to be the set of cells A = {(¿, 1), (i + 1,2),..., (i — 1, n)} where all arithmetic is performed in Zn (using the reduced residues {1,2,..., n}). We say that the diagonals A and Di+1 are consecutive diagonals. We should note that in [3] there is a construction of a shiftable H(n; 4) for all n > 4 that uses 4 consecutive diagonals. All the constructions in this paper are based on filling in the cells of a fixed collection of diagonals. To aid in these constructions we define the following procedure for filling a sequence of cells on a diagonal. It is termed diag and it has six parameters. In an n x n array A the procedure diag (r, c, s, A1, A2, installs the entries A[r + iA1, c + iA1] = s + iA2 for i = 0,1,... — 1. Here all arithmetic on the row and column indices is performed modulo n, where the set of reduced residues is {1, 2,..., n}. The following summarizes the parameters used in the diag procedure: • r denotes the starting row, • c denotes the starting column, • s denotes the starting symbol, • A1 denotes how much the row and column are changed at each step, • A2 denotes how much the symbol is changed at each step, and • I is the length of the chain. The following example shows the use of the above definition and is also an example of the construction which will be described in Theorem 2.2. Example 2.1. A shiftable H(11; 4) where the filled cells are contained in two sets of consecutive diagonals. 84 Ars Math. Contemp. 13 (2017) 107-123 The Heffter array H(11; 4) below is constructed via the following procedures: diag(4,1,1,1, 2, 11); diag(5,1, -2,1, -2,11); diag(4, 7, -23,1, -2,11); diag(5, 7, 24,1, 2,11). 38 -39 -16 17 40 -41 -18 19 42 -43 -20 21 1 44 -23 -22 -2 3 24 -25 -4 5 26 -27 -6 7 28 -29 -8 9 30 -31 -33 -10 11 32 34 -35 -12 13 36 -37 -14 15 We point out a few properties of the Heffter array in Example 2.1 which will be useful in the proof of the main theorem of this section. First we note that all of the filled cells are in the two pairs of consecutive diagonals {D4, D5} and {D9, D10} and that the sum of the symbols in the columns of one of the pairs of diagonals is +1 while the other adds to -1. Hence every column adds to 0. The rows are similar except for row 4. In this row the sum of the symbols in D4 and D5 is -21 while the sum of the symbols in D9 and D10 is +21. So all the row sums are 0. It is also apparent that each row and each column contain two positive values and two negative values making this a shiftable array. Finally it is clear that the support of D4 and D5 is {1,2,..., 22} while the support of D9 and D10 is {23,24,..., 44}. We have thus shown that this is indeed a shiftable integer H(11; 4) where all the filled cells are in the two pairs of consecutive diagonals. The following is the main theorem of this section. Theorem 2.2. For every n > 4 and any two disjoint pairs of consecutive diagonals, there exists a shiftable integer Heffter array H(n; 4) with filled cells contained in the four diagonals. Proof. Assume that the two pairs of consecutive diagonals are {Da,Da+1} and {Db, Db+1} with b > a +1. We define the square H using the diag procedures as in Example 2.1 above. So let H be constructed from diag (a, 1,1,1, 2, n), diag (a + 1,1, -2,1, -2, n), diag (a, n + a - b +1, -2n - 1,1, -2, n), and diag (a + 1, n + a - b +1, 2n + 2,1, 2, n). Clearly diagonal Da is filled from the procedure diag (a, 1,1,1,2, n) while diagonal Da+1 is filled from diag (a + 1,1, -2,1, -2, n). We next note that a cell (i, j) gets filled J. H. Dinitz and I. M. Wanless: The existence of square integer Heffter arrays 85 from the procedure diag (a, n + a — b +1, —2n — 1,1, -2, n) if and only if j — i = (n + a — b +1) — a = n — b +1 = 1 — b (mod n). So cell (b, 1) is filled from this procedure. Since I = n in this procedure we have that every cell in Db is filled. Similarly every cell in Db+1 is filled from the procedure diag (a + 1, n + a — b +1, 2n + 2,1,2, n). Considering the column sums, we see that in each column the sum of the cells in Da and Da+1 is —1, while the sum of the cells in Db and Db+1 is +1. Hence the sum of the symbols in each column is 0. Similarly, if r = a, then the sum of the cells in row r in Da and Da+1 is +1, while the sum of the cells in row r in Db and Db+1 is —1. So the sum of the symbols in each row r = a is 0. Now consider row a. The symbols from Da,Da+1, Db and Db+1 are 1, — 2n, —2n — 1 and 4n, respectively, and so the symbols in this row also add to 0. We next check the support of H. We see that the support of diag (a, 1,1,1,2, n) is {1, 3,..., 2n — 1}, while the support of diag (a + 1,1, —2,1, —2, n) is {2,4,..., 2n} so together they cover the symbols {1, 2,..., 2n}. Further, we have that the support of diag (a, n + a — b +1, —2n — 1,1, —2, n) is {2n +1, 2n + 3,..., 4n — 1}, while the support of diag (a + 1, n + a — b + 1, 2n + 2,1, 2, n) is {2n + 2, 2n + 4,..., 4n}, so these two diagonals cover the symbols {2n +1, 2n + 2,..., 4n}. Hence the support of H is the required {1, 2,..., 4n}. Finally it is clear from the construction that each row and each column contains two positive numbers and two negative numbers. Thus we have shown that H is indeed a shiftable integer H(n; 4), as desired. □ 3 H(n ;k) when n = 3 (mod 4) and k = 1 (mod 4) In this section we first give a direct construction for H(n; 5) with n = 3 (mod 4) where all of the filled cells are on exactly 5 diagonals. We then use Theorem 2.2 repeatedly to construct H(n; k) for all n = 3 (mod 4) with n > 7, and all k = 1 (mod 4) with 5 < k < n — 2. We begin with an example of the main construction of this section. Hopefully, the reader can see the type of patterns which will exist in the general case. Example 3.1. An H(11,5). 10 53 24 —33 —54 —36 —9 44 32 —31 —45 —8 52 30 —29 —37 —7 43 28 —27 —46 —6 51 26 —25 —21 —38 — 11 47 23 12 — 13 —42 4 39 14 — 15 —50 3 48 16 — 17 —41 2 40 18 — 19 —49 —5 55 35 20 —22 —34 1 Theorem 3.2. There exists an H(n, 5) for all n = 3 (mod 4) with n > 7. Proof. Let h = (n + 1)/2 and q = (n — 3)/4. We construct an n x n array H using the 86 Ars Math. Contemp. 13 (2017) 107-123 following procedures. The procedures are labeled A to N. A diag(h + 1,h + 1,h - 2,1,-1, h - 3); B diag(3, 3, -(n — 3), 1, 1, h — 3); C diag(2, 3, 4n, 2,-1, q); D diag(3, 2,-(4n + 1), 2,-1,q); E diag(3, 4, 5n - 3, 2, -1,q); F diag(4, 3,-(3n + 4), 2,-1,q); G diag(h +1,h, -(4n - q), 2,1,q); H diag(h + 2, h + 1,-(5n - q - 3), 2,1,q); I diag(h, h + 1, 4n + q + 1, 2,1,q); J diag(h +1,h + 2, 3n + q + 4, 2, 1,q); K diag(h + 1, 2, -(n + 2), 1, -2, h - 2); L diag(h + 1,1,n + 1, 1, 2, h - 1); M diag(1, h + 1, -3n, 1, 2, h - 1); N diag(2, h + 1, 3n - 1,1, -2, h - 2); We also fill the following cells in an ad hoc manner. H [1,1] = n - 1; H[1, n] = -(5n - 1); H[h, 1] = -(2n - 1); H[n - 1, n - 1] = -(h - 1); H [n, h] = -2n; H [1, 2] = 5n - 2; H [2,1] = - (3n + 3); H[h, h] = -n; H [n - 1, n] = 5n; H[n, n - 1] = -(3n + 1); H [1,h] = 2n + 2; H[2, 2] = -(n - 2); H[h, n] = 2n + 1; H [n, 1] = 3n + 2; H [n, n] = 1. We now prove that the array constructed by the description above is indeed an integer H(n; 5). To aid in the proof we give a schematic picture of where each of the diagonal procedures fills cells (see Figure 1). The first cell in each of these procedures is shaded and we have placed an X in the ad hoc cells. (In this picture we used n = 15, so h = 8 and q = 3.) We first check that the rows all add to 0 (in the integers). Row 1: There are four ad hoc values plus the first value in diagonal M. The sum is (n - 1) + (5n - 2) + (2n + 2) - 3n - (5n - 1) = 0. Row 2: The sum is -(3n + 3) - (n - 2) + 4n + (3n - 1) - (3n - 2) = 0. Rows 3 to h - 1: First notice that in all of these rows the sum of the N and the M diagonal cells is +1 so we must show that the sum of the three cells in the three center diagonals is -1. There are two cases depending on whether the row r is odd or even. If r is odd, then write r = 3 + 2k where 0 < k < q - 1. Notice that from the D, B and E diagonal cells we get the following sum: -(4n +1) - k - (n - 3) + J. H. Dinitz and I. M. Wanless: The existence of square integer Heffter arrays 87 X X X M X X X C N M D B E N M F B C N M D B E N M F B C N M D B E N M X F X I X L K G A J L K H A I L K G A J L K H A I L K G A J L K H X X X L X X X Figure 1: Construction in Theorem 3.2 for n = 15. 2k + (5n — 3) — k = —1 as desired. If r is even, then write r = 4 + 2k where 0 < k < q — 2. From the F, B and C diagonal cells we get the following sum: — (3n + 4) — k — (n — 3) + 1 + 2k + 4n — 1 — k = —1, as desired. Row h: There are three ad hoc values plus the last of the F diagonal as well as the first of the I diagonal. We get the row sum: — (2n — 1) — n + (2n + 1) — (3n + 4) — (q — 1) +4n + q +1=0. Rows h+1 to n — 2: Note that in all of these rows the sum of the L and the K diagonal cells is — 1 so we must show that the sum of the three cells in the three center diagonals is +1. There are again two cases depending on whether the row r is odd or even. If r is odd, noting that h is even, we write r = (h + 1) + 2k where 0 < k < q — 1. Now, from the G, A and J diagonal cells we get the following sum: — (4n — q) + k + (h — 2) — 2k +(3n+q+4) + k = —n+2q + h+2 = 1. If r is even, writer = (h + 2)+2k where 0 < k < q — 2. From the H, A and I diagonal cells we get the following sum: — (5n — q — 3) + k + (h — 3) — 2k + (4n + q + 2) + k = —n + 2q + h + 2=1. Row n — 1 : We add the values in diagonals L, K and H with two ad hoc values to get: (n + 1) + 2(h — 3) — (n + 2) — 2(h — 3) — (5n — q — 3) + (q — 1) — (h — 1) + 5n = —h + 2q + 2 = 0. Rown: Thesumis (3n + 2) — 2n — (3n +1) + 1 + (n +1) + 2(h — 2) = —n + 2h — 1 = 0. So all rows add to zero. Next we check that the columns also all add to zero. Column 1: There are four ad hoc values plus the first value in diagonal L. The sum is (n — 1) — (3n + 3) — (2n — 1) + (n + 1) + (3n + 2) = 0. Column 2: The sum is 5n — 2 — (n — 2) — (4n +1) — (n + 2) + (n + 1) + 2 = 0. 88 Ars Math. Contemp. 13 (2017) 107-123 Columns 3 to h — 1: Note that in all of these columns the sum of the L and the K diagonal cells is +1 so we must show that the sum of the three cells in the three center diagonals is —1. There are two cases depending on whether the column c is odd or even. If c is odd, then write c = 3 + 2k where 0 < k < q — 1. From the C, B and F diagonal cells we get the following sum: 4n — k — (n — 3) + 2k — (3n + 4) — k = —1. If c is even, then write c = 4 + 2k where 0 < k < q — 2. From the E, B and D diagonal cells we get the following sum: (5n — 3) — k — (n — 4) + 2k — (4n +1) — 1 — k = —1, as desired. Column h: There are three ad hoc values plus the last of the E diagonal as well as the first of the G diagonal. We get (2n + 2) — n — 2n + (5n — 3) — (q — 1) — (4n — q) = 0. Columns h + 1 to n — 2: In all of these columns the sum of the M and the N diagonal cells is —1, so we must show that the sum of the three cells in the three center diagonals is +1. There are again two cases depending on whether the column c is odd or even. If c is odd, noting that h is even, we write c = (h + 1) + 2k where 0 < k < q — 1. Now, from the I, A and H diagonal cells we get the following sum: (4n + q + 1) + k + (h — 2) — 2k — (5n — q — 3) + k = —n + 2q + h + 2 = 1. If c is even, write c = (h + 2) + 2k where 0 < k < q — 2. From the J, A and G diagonal cells we get the following sum: (3n + q + 4)+ k + (h — 3) — 2k — (4n — q) +1 + k = —n + 2q + h + 2=1. Column n — 1 : We add the values in diagonals M, N and J with two ad hoc values to get: (—3n) + 2(h — 3) + (3n — 1) — 2(h — 3) + (3n + q + 4)+ q — 1 — (h — 1) — (3n + 1) = 2q — h + 2 = 0. Column n: The sum is —5n +1 — 3n + 2(h — 2) + 2n +1 + 5n +1 = —n + 2h — 1 = 0. So we have shown that all column sums are zero. Next we consider the support of H. We do this by looking at the elements used in each of the diagonals as well as the ad hoc symbols. We will write [u, v](w) if the elements in diagonal W consist of the integers in the closed interval [u, v] and we give the ad hoc symbols individually. Note that we write all the numbers in terms of the value q (where 4q + 3 = n). The support of H is: {1, [2, 2q](A), 2q + 1, [2q + 2, 4q](B), 4q +1, 4q + 2, 4q + 3, [4q + 4, 8q + 4](K U L), 8q + 5, 8q + 6, 8q + 7, 8q + 8, [8q + 9,12q + 9](m U n), 12q + 10,12q + 11, 12q + 12, [12q + 13, 13q + 12](f), [13q + 13,14q + 12](j), [14q + 13, 15q + 12](g), [15q + 13, 16q + 12](c), [16q + 13, 17q + 12](d), [17q + 13, 18q + 12](i), [18q + 13,19q + 12](h), [19q + 13, 20q + 12](e), 20q + 13, 20q + 14, 20q + 15} = [1, 20q +15] = [1, 5n]. We have shown that H is indeed an integer Heffter array H(n; 5). □ We are now ready to prove the main theorem of this section. Let k = 5 + 4s. To construct an H(n; k) we start with the H(n; 5) constructed in Theorem 3.2 and add s disjoint H(n; 4) (with the symbols shifted accordingly), as constructed in Theorem 2.2. The details are given in the following theorem. Theorem 3.3. There exists an integer Heffter array H(n; k) for all n = 3 (mod 4) and k = 1 (mod 4) with n ^ 7 and 5 ^ k ^ n — 2. J. H. Dinitz and I. M. Wanless: The existence of square integer Heffter arrays 89 Proof. Again let h = (n + 1)/2, noting that h is necessarily even, and let k = 5 + 4s. When s = 0 we are done by Theorem 3.2. So we assume that s > 1, and hence that 4 < 4s < n—7. Begin with H = H(n; 5) constructed in Theorem 3.2. We place s (shifted) H(n; 4) constructed in Theorem 2.2 in 4s empty diagonals of H. These empty diagonals will come in pairs of consecutive diagonals. Specifically, for each 0 < t < s — 1 place Ht = H(n; 4) ± (5n + 4nt) on the 4 diagonals £3+24, £4+24, Dh+2+2t, and Dh+3+2t. A few things need to be checked. The filled diagonals in H are Di, D2, Dh, Dh+i, and Dn. The diagonals that get filled with the Ht's are D3, D4,..., D1+2s, D2+2s and Dh+2, Dh+3,..., Dh+2s, Dh+2s+1. Since 4s < n — 7, then 2s + 2 < h — 2 and also h + 2s + 1 < n — 2. So the filled diagonals in H, H1, H2,..., Hs are all disjoint. The row and column sums in H as well as in each Ht, 0 < t < s — 1 is zero, hence the resulting array has row and column sum zero. Finally, note that the support of H is [1, 5n] and for each Ht the support is [5n + 4nt + 1, 5n + 4nt + 4n] = [5n + 4nt + 1, 9n + 4nt]. So the support in the final array is s — 1 [1, 5n] U (J [5n + 4nt +1, 9n + 4nt] t=0 = [1, 5n] U [5n + 1, 9n] U [9n + 1, 13n] U • • • U [5n + 4n(s — 1) + 1, 9n + 4n(s — 1)]. Since 9n + 4n(s — 1) = n(5 + 4s) = nk, the support is [1,nk], completing the proof. □ 4 H(n ;k) when n = 0 (mod 4) and k = 1 (mod 4) This section follows the same structure as Section 3. We first give a direct construction for H(n; 5) with n = 0 (mod 4) where all of the filled cells are on exactly 5 diagonals. We then use Theorem 2.2 repeatedly to construct H(n; k) for all n = 0 (mod 4) with n > 8, and all k = 1 (mod 4) with 5 < k < n — 3. We again begin with an example of the main construction of this section. Example 4.1. An H(16,5). 46 — 14 62 —49 —45 —63 — 13 75 44 —43 —50 — 12 61 42 —41 —64 — 11 74 40 —39 —51 — 10 60 38 —37 —65 —9 73 36 —35 —52 —8 59 34 —33 —30 —66 77 67 —48 15 — 16 —58 6 53 17 — 18 —72 5 68 19 —20 —57 4 54 21 —22 —71 3 69 23 —24 —56 2 55 25 —26 —70 —7 78 —32 27 80 —47 —28 1 76 —79 —29 31 90 Ars Math. Contemp. 13 (2017) 107-123 Theorem 4.2. There exists an H(n, 5) for all n = 0 (mod 4) with n > 8. Proof. Let h = n/2 and q = n/4. We construct an n x n array H using the following procedures. The procedures are labeled A to N. A diag(h +1,h + 2, h - 2,1,-1, h - 3); B diag(1, 2,-(n - 2), 1, 1, h - 1); C diag(1, 3, 4n - 2, 2, -1,q); D diag(2, 2,-(4n - 1), 2,-1,q); E diag(2, 4, 5n - 5, 2, -1,q - 1); F diag(3, 3,-(3n + 2), 2,-1,q - 1); G diag(h + 1, h + 1, -(4n - q - 2), 2,1, q - 1); H diag(h + 2, h + 2, -(5n - q - 4), 2,1, q - 1); I diag(h, h + 2, 4n + q - 1, 2,1,q - 1); J diag(h +1,h + 3, 3n + q + 1, 2, 1,q - 1); K diag(h + 1,1,n - 1, 1, 2, h - 1); L diag(h + 1, 2, -n, 1, -2, h - 2); M diag(2, h + 2, 3n - 4,1, -2, h - 2); N diag(1, h + 2, -(3n - 3), 1, 2, h - 1); We also fill the following cells in an ad hoc manner. H[1,1] = 3n - 2; H[h, 1] = -(2n - 2); H[h, n] = -3n; H[n - 2, n] = 5n - 2; H[n - 1, h] = 5n; H[n - 1, n] = -(2n - 4); H[n, 2] = 5n - 4; H[n, h + 1] = -(2n - 3); H [1,h +1 H[h, h + 1 H[n - 2, n - 1 H [n - 1,1 H [n - 1, n - 1 H [n, 1 H[n, h H[n, n -(3n +1); 5n - 3; -(h -1); -2n; -(3n - 1); 1; -(5n - 1); 2n — 1. We now prove that the array constructed by the description above is indeed an integer H(n; 5). To aid in the proof we again give a schematic picture of where each of the diagonal procedures fills cells (see Figure 2). The first cell in each of these procedures is shaded and we have placed an X in the ad hoc cells. (In this picture we used n = 16, so h = 8 and q = 4.) We first check that the rows all add to zero. Row 1: There are two ad hoc values plus the first value in diagonals B, C and N. The sum is (3n - 2) - (n - 2) + (4n - 2) - (3n +1) - (3n - 3) = 0. J. H. Dinitz and I. M. Wanless: The existence of square integer Heffter arrays 91 X B C X N D B E M N F B C M N D B E M N F B C M N D B E M N F B C M N X D X I X K L G A J K L H A I K L G A J K L H A I K L G A J K L H X X X K X X X X X X X X Figure 2: Construction in Theorem 4.2 for n = 16. Rows 2 to h — 1: In all of these rows the sum of the N and the M diagonal cells is +1 so we must show that the sum of the three cells in the three center diagonals is — 1. There are two cases depending on whether the row r is odd or even. If r is even, then write r = 2 + 2k where 0 < k < q — 2. From the D, B and E diagonal cells we get the following sum: —(4n — 1) — k — (n — 3) + 2k + (5n — 5) — k = —1. If r is odd, then write r = 3 + 2k where 0 < k < q — 2. Notice that from the F, B and C diagonal cells we get the following sum: — (3n + 2) — k — (n — 4) + 2k + (4n — 3) — k = — 1 as desired. Row h: The sum is: —(2n — 2) — (4n — 1) — (q — 1) + (5n — 3) + (4n + q — 1) — 3n = 0. Row h + 1 to n — 3: Note that in all of these rows the sum of the L and the K diagonal cells is — 1 so we must show that the sum of the three cells in the three center diagonals is +1. There are again two cases depending on whether the row r is odd or even. If r is odd, noting that h is even, we write r = (h + 1) + 2k where 0 < k < q — 2. Now, from the G, A and J diagonal cells we get the following sum: — (4n — q — 2) + k + (h — 2) — 2k + (3n + q + 1) + k = —n + 2q + h +1 = 1, as desired. If r is even, write r = (h + 2) + 2k where 0 < k < q — 3. From the H, A and I diagonal cells we get the following sum: —(5n —q — 4)+k+(h —3)—2k+(4n+q) + k = —n+2q+h+1 = 1. Row n — 2: The sum is: (n — 1) + 2(h — 3) — n — 2(h — 3) — (5n — q — 4) + (q — 2) — h + 1 + (5n — 2) = 2q — h = 0. Rown—1: Thesumis: —2n+(n—1)+2(h—2)+5n—(3n —1) —(2n—4) = —n+2h = 0. Row n: The sum is: 1 + (5n — 4) — (5n — 1) — (2n — 3) + (2n — 1) = 0. So all rows add to zero. Next we check that the columns also all add to zero. Column 1: There are four ad hoc values plus the first value in diagonal K. The sum is (3n — 2) — (2n — 2) + (n — 1) — 2n +1 = 0. 92 Ars Math. Contemp. 13 (2017) 107-123 Column 2: The sum is: -(n - 2) - (4n - 1) - n + (n + 1) + (5n - 4) = 0. Columns 3 to h - 1: Note that in all of these columns the sum of the L and the K diagonal cells is +1, so we must show that the sum of the three cells in the three center diagonals is -1. There are two cases depending on whether the column c is odd or even. If c is odd, then write c = 3 + 2k where 0 < k < q - 2. From the C, B and F diagonal cells we get the following sum: (4n - 2) - k - (n - 3) + 2k - (3n + 2) - k = -1. If c is even, then write c = 4 + 2k where 0 < k < q - 3. From the E, B and D diagonal cells we get the following sum: (5n - 5) - k - (n - 4)+2k - 4n - k = -1, as desired. Column h: There are two ad hoc values plus the last of the E, B and D diagonals. We get 5n - (5n - 1) + (5n - 5) - (q - 2) - (n - 2) + (h - 2) - (4n - 1) - (q - 1) = 0. Column h +1: The sum is: -(3n + 1) + (4n - 2) - (q - 1) + (5n - 3) - (4n - q -2) - (2n - 3) = 0. Columns h + 2 to n - 2: In all of these columns the sum of the M and the N diagonal cells is -1, so we must show that the sum of the three cells in the three center diagonals is +1. There are again two cases depending on whether the column c is odd or even. If c is even, noting that h is even, write c = (h + 2) + 2k where 0 < k < q - 2. From the I, A and H diagonal cells we get the following sum: (4n + q - 1) + k + (h - 2) - 2k - (5n - q - 4)+ k = -n + h + 2q +1 = 1. If c is odd, we write c = (h + 3) + 2k where 0 < k < q - 3. Now, from the J, A and G diagonal cells we get the following sum: (3n + q +1) + k + (h - 3) - 2k - (4n - q - 3) + k = -n + 2q + h +1 = 1. Column n - 1: The sum is: -(3n - 3) + 2(h - 3) + (3n - 4) - 2(h - 3) + (3n + q + 1) + (q - 2) - (h - 1) - (3n - 1) = 2q - h = 0. Column n: The sum is: -(3n - 3) + 2(h - 2) - 3n + (5n - 2) - (2n - 4) + (2n - 1) = -n + 2h = 0. So we have shown that all column sums are zero. Next we consider the support of H. We do this by looking at the elements used in each of the diagonals as well as the ad hoc symbols used. We again write [u, v](w) if the elements in diagonal W consist of the integers in the closed interval [u, v] and we give the ad hoc symbols individually. Note that we write all the numbers in terms of the value q (where 4q = n). The support of H is: {1, [2, 2q - 2](A), 2q - 1, [2q, 4q - 2](b), [4q - 1, 8q - 5](K U l), 8q - 4, 8q - 3, 8q - 2, 8q - 1, 8q, [8q + 1,12q - 3](m U n), 12q - 2, 12q - 1,12q, 12q + 1, [12q + 2,13q](F), [13q + 1, 14q - 1](j), [14q, 15q - 2](g), [15q - 1, 16q - 2](c), [16q - 1,17q - 2](d), [17q - 1,18q - 3](i), [18q - 2,19q - 4](h), [19q - 3, 20q - 5](e), 20q - 4, 20q - 3, 20q - 2, 20q - 1, 20q} = [1, 20q] = [1, 5n]. We have shown that the array H is indeed an integer Heffter array H(n; 5). □ We now present the main theorem of this section. Theorem 4.3. There exists an integer Heffter array H(n; k) for all n = 0 (mod 4) and k = 1 (mod 4) with 5 ^ k ^ n - 3. J. H. Dinitz and I. M. Wanless: The existence of square integer Heffter arrays 93 Proof. The proof is very similar to that of Theorem 3.3. As above, let h = n/2, where h is necessarily even, and let k = 5 + 4s. When s = 0 we are done by Theorem 4.2. So we assume that s > 1, and hence that 4 < 4s < n - 8. Begin with H = H(n; 5) constructed in Theorem 4.2. We place s (shifted) H(n; 4) constructed in Theorem 2.2 in 4s empty diagonals of the H(n; 5). These empty diagonals will again come in pairs of consecutive diagonals. Specifically, for each 0 < t < s - 1 place Ht = H(n; 4) ± (5n + 4nt) on the four diagonals D2+21, D3+1, Dh+2+21, and Dh+3+2t. A few things need to be checked. The filled diagonals in H are D^ Dh, Dh+1, Dn-1, and Dn. The diagonals that get filled with the Ht's are D2, D3,..., D2s, D1+2s and Dh+2, Dh+3, ..., Dh+2s, Dh+2s+1. Since 4s < n - 8, then 2s + 1 < h - 3 and also h + 2s + 1 < n - 3. So the filled diagonals in H, H1, H2,..., Hs are all disjoint. The row and column sums in H as well as in each Ht, 0 < t < s - 1 is zero, hence the resulting array has row and column sum zero. Finally, note that the support of H is [1, 5n] and for each Ht the support is [5n + 4nt + 1, 5n + 4nt + 4n] = [5n + 4nt + 1, 9n + 4nt]. So the support in the final array is s — 1 [1, 5n] U (J [5n + 4nt +1, 9n + 4nt] t=0 = [1, 5n] U [5n + 1, 9n] U [9n + 1, 13n] U • • • U [5n + 4n(s - 1) + 1, 9n + 4n(s - 1)]. Since 9n + 4n(s - 1) = n(5 + 4s) = nk, the support is [1,nk], completing the proof. □ 5 Conclusion In the paper [3], it was proven that the necessary conditions for the existence of an integer H(n; k) are that n > k and nk = 0,3 (mod 4). Furthermore, this condition was proved to be sufficient except possibly when n = 0 or 3 (mod 4) and k = 1 (mod 4). In Section 3 we proved that H(n; k) exist when n = 3 (mod 4) and k = 1 (mod 4) and in Section 4 we proved that H(n; k) exist when n = 0 (mod 4) and k = 1 (mod 4). From this we have the main result of this paper. Theorem 5.1. There exists an integer Heffter array H (n; k) if and only if 3 ^ k ^ n and nk = 0, 3 (mod 4). In future work we will consider the case when the Heffter array H(n; k) is not an integer Heffter array. In this case the only necessary condition is that 3 < k < n. References [1] D. S. Archdeacon, Heffter arrays and biembedding graphs on surfaces, Electron. J. Combin. 22 (2015), #P1.74. [2] D. S. Archdeacon, T. Boothby and J. H. Dinitz, Tight Heffter arrays exist for all possible values, J. Combin. Des. 25 (2017), 5-35, doi:10.1002/jcd.21520. [3] D. S. Archdeacon, J. H. Dinitz, D. M. Donovan and E. S. Yazici, Square integer Heffter arrays with empty cells, Des. Codes Cryptogr. 77 (2015), 409-426, doi:10.1007/s10623-015-0076-4. [4] D. S. Archdeacon, J. H. Dinitz, A. Mattern and D. R. Stinson, On partial sums in cyclic groups, J. Combin. Math. Combin. Comput. 98 (2016), 327-342. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 95-106 3-pyramidal Steiner triple systems * Marco Buratti Dipartimento di Matematica e Informática, Universitá di Perugia, via Vanvitelli 1, 06123 Perugia, Italy Gloria Rinaldi Dipartimento di Scienze e Metodi dell'Ingegneria, Universitä di Modena e Reggio Emilia, via Amendola 2, 42122 Reggio Emilia, Italy Tommaso Traetta Department of Mathematics, Ryerson University, Toronto (ON) M5B 2K3, Canada Received 21 December 2015, accepted 30 March 2016, published online 19 February 2017 A design is said to be f -pyramidal when it has an automorphism group which fixes f points and acts sharply transitively on all the others. The problem of establishing the set of values of v for which there exists an f -pyramidal Steiner triple system of order v has been deeply investigated in the case f = 1 but it remains open for a special class of values of v. The same problem for the next possible f, which is f = 3, is here completely solved: there exists a 3-pyramidal Steiner triple system of order v if and only if v = 7,9,15 (mod 24) or v = 3,19 (mod 48). Keywords: Steiner triple system, group action, difference family, Skolem sequence, Langford sequence. Math. Subj. Class.: 51E10, 20B25, 05B07, 05B10 1 Introduction A Steiner triple system of order v, briefly STS(v), is a pair (V, B) where V is a set of v points and B is a set of 3-subsets (blocks or triples) of V with the property that any two distinct points are contained in exactly one block. Apart from the trivial case v = 0 in * Research performed within the activity of INdAM-GNSAGA with the financial support of the italian Ministry MIUR, project "Combinatorial Designs, Graphs and their Applications". E-mail addresses: buratti@dmi.unipg.it (Marco Buratti), gloria.rinaldi@unimore.it (Gloria Rinaldi), tommaso.traetta@ryerson.ca, traetta.tommaso@gmail.com (Tommaso Traetta) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract 96 Ars Math. Contemp. 13 (2017) 107-123 which both V and B are empty, it is well known that a STS(v) exists if and only if v = 1 or 3 (mod 6). For general background on STSs we refer to [7]. Steiner triple systems having an automorphism with a prescribed property or an automorphism group with a prescribed action have drawn much attention since a long time. It was proved by Peltesohn [15] that a STS(v) with an automorphism cyclically permuting all points, briefly a cyclic STS(v), exists for any possible v but v = 9. The existence question for a STS(v) with an involutory automorphism fixing exactly one point, briefly a reverse STS(v), has been settled by means of three different contributions of Doyen [9], Rosa [18] and Teirlinck [21]; it exists if and only if v = 1,3,9,19 (mod 24). In [16] Phelps and Rosa proved that there exists a STS(v) with an automorphism cyclically permuting all but one point, briefly a 1-rotational STS(v), if and only if v = 3 or 9 (mod 24). Note that a cyclic or 1-rotational STS may be viewed as a STS with a cyclic automorphism group acting sharply transitively on all points or all but one point, respectively. Thus one may ask, more generally, for a STS with an automorphism group G having the same kind of action without the request that G be cyclic. Speaking of a regular STS(v) we mean a STS(v) for which there is at least one group G acting sharply transitively on the points. Also, speaking of a 1-rotational STS(v) we mean a STS(v) with at least one automorphism group acting regularly on all but one points. The only STS(9), which is the point-line design associated with the affine plane over Z3, is clearly regular under Z3. Thus, in view of Peltesohn's result, there exists a regular STS(v) for any admissible v. The problem of determining the set of values of v for which there exists a 1-rotational STS(v) (under some group) has been deeply investigated in [2, 4]. Such a STS is necessarily reverse so that v must be congruent to 1, 3, 9, or 19 (mod 24). For the cases v = 3 or 9 (mod 24) the existence clearly follows from the result by Phelps and Rosa. In the case v = 19 (mod 24) we do not have existence only when v = 6PQ + 1 with P = 1 or a product of pairwise distinct primes congruent to 5 (mod 12) and with Q a product of an odd number of pairwise distinct primes congruent to 11 (mod 12). The most difficult case is v = 1 (mod 24) where the existence remains still uncertain only when all the following conditions are simultaneously satisfied: v = (p3 - p)n +1 = 1 (mod 96) with p a prime; n = 0 (mod 4); the odd part of v - 1 is square-free and without prime factors = 1 (mod 6). Of course one might consider the more specific problems of determining all groups G for which there exists a STS which is regular under G and all groups G for which there exists a STS which is 1-rotational under G. These problems appear at this moment quite hard. The same problems can be relaxed by asking for which v there is a STS(v) which is regular (1-rotational) under a group belonging to an assigned general class. For instance, the first author proved that there exists a STS(v) which is 1-rotational under some abelian group if and only if either v = 3,9 (mod 24) or v = 1,19 (mod 72). Also, Mishima [14] proved that there exists a STS(v) which is 1-rotational under a dicyclic group if and only if v = 9 (mod 24). We have quoted only the results which are closer to the problem that we are going to study in this paper. Indeed the literature on Steiner triple systems and their automorphism groups is quite wide. For example, results concerning the full automorphism group of a STS have been obtained by Mendelsohn [13] and Lovegrove [12]. Now we want to consider the problem of determining the set of values of v for which there exists a STS(v) with an automorphism group fixing f points and acting sharply transitively on the other v - f points. Such a STS will be called f -pyramidal. Of course the M. Buratti el al.: 3-pyramidal Steiner triple systems 97 cases f = 0 and f = 1 correspond, respectively, to the regular and 1-rotational STSs discussed above. It is natural to study the next possible case that is f = 3. This is because, as we are going to see in the next lemma, the fixed points of an f -pyramidal STS(v) form a subsystem of order f so that, for f = 0, we have f = 1 or 3 (mod 6); also, if f = v, then f < v/2. Lemma 1.1. A necessary condition for the existence of an f -pyramidal STS(v) is that f = 0 or f = 1, 3 (mod 6), and f = v or f < v/2. Proof. Let (V, B) be an f-pyramidal STS(v) under the action of a group G. Assume, w.l.o.g., that G is additive and let F = j^i,..., to/} be the set of points fixed by G. Obviously G has order v - f, the set of points V can be identified with F U G, and the action of G on V can be identified with the addition on the right with the assumption that TOj + g = TOj for each rxj g F and each g g G. If a block B g B contains two distinct fixed points, say x>j and TOj, then B + g = B for every g g G otherwise B would have two distinct blocks, B and B + g, passing through the two points Wj and TOj. So, the third vertex of B is also fixed by G. It easily follows that all blocks of B contained in F form a STS(f) so that we have f = 0 or f = 1,3 (mod 6), and f = v or f < v/2. □ The main result of this paper is a complete solution to the existence problem for a 3-pyramidal STS(v). Theorem 1.2. There exists a 3-pyramidal STS(v) if and only if v = 7, 9, 15 (mod 24) or v = 3, 19 (mod 48). The "if part" of this theorem will be proved in Section 3 which therefore will give non-existence results: for v = 1, 13, 21 (mod 24) or v = 27, 43 (mod 48) there is no 3-pyramidal STS(v). The "only if part" will be proved in Section 4 where we will give an explicit construction of a 3-pyramidal STS (v) whenever v = 7,9,15 (mod 24) or v = 3,19 (mod 48). First, in the next section, we have to translate our problem into algebraic terms: any f -pyramidal STS is completely equivalent to a suitable difference family. 2 Difference families and pyramidal STSs As a natural generalization of the concept of a relative difference set [17], the first author introduced [3] difference families in a group G relative to a subgroup of G or, even more generally [5], relative to a partial spread of G. By a partial spread of a group G one means a set E of subgroups of G whose mutual intersections are all trivial. One omits the attribute "partial" in the special case that the subgroups of E cover all G. Let E be a partial spread of an additively written group G, let F be a set of k-subsets G, and let AF be the list of all possible differences x - y with (x, y) an ordered pair of distinct elements of a member of F. One says that F is a (G, E, k, 1)-difference family (DF) if every group element appears 0 or 1 times in AF according to whether it belongs or does not belong to some member of E, respectively. We say that E is of type jdi1,..., dn } if this is the multiset (written in "exponential" notation) of the orders of all subgroups belonging to E and we speak of a (G, jdi1 ,...,dn }, k, 1)-DF. It is obvious that any STS(v) is v-pyramidal under the trivial group. The following theorem explains how to construct an f-pyramidal STS(v) with f < v/2. It generalizes Theorem 1.1 in [3] which corresponds to the case f =1. 98 Ars Math. Contemp. 13 (2017) 107-123 Theorem 2.1. There exists an f -pyramidal STS(v) with f < v/2 if and only if there exists a (G, {2f, 3e}, 3,1)-DF for a suitable group G of order v — f with exactly f involutions, and a suitable integer e. Proof. (=^). Let (V, B) be an f-pyramidal STS(v) under an additive group G. We can assume that V = F U G with F and the action of G on V defined as in Lemma 1.1. For 1 < i < f, let Bi = {ra^ 0, xj} be the block of B containing the points wi and 0. We have Bi — xi = {wi, —xi, 0} so that both Bi and Bi — xi contain the points wi and 0. It necessarily follows that Bi — xi = Bi; hence —xi = xi which means that xi is an involution. Conversely, if x is an involution of G and B = {0, x, y} is the block through 0 and x, then B + x = {x, 0, y + x} would also contain 0 and x so that B + x = B. This means that y + x = y and this is possible only if y G F. Hence y = wi and x = xi for a suitable i. We conclude that {x^..., xf } is the set of all involutions of G. Let F be a complete system of representatives for the G-orbits on the blocks of B with trivial G-stabilizer. Reasoning as in the "if part" of Theorem 2.2 in [4], one can see that F is a (G, E, 3,1)-DF where E is the partial spread of G consisting of all 2-subgroups {0, xi} (i = 1,..., f) of G and all 3-subgroups of G belonging to B. (^=). Now assume that f = 1 or 3 (mod 6) and that F is a (G, E, 3,1)-DF with G a group of order v — f having exactly f involutions and with E a partial spread of G of type {2f, 3e}. Take an f-set F = {w^ ..., Wf} disjoint with G and let (F, BTO) be any STS(f) (which exists because we assumed that f = 1 or 3 (mod 6)). For i = 2, 3, let Ei be the set of subgroups of order i belonging to E. Set E2 = {Si | 1 < i < f} and E+ = {Si U {wi} | 1 < i < f}. Then, as in the "only if part" of Theorem 2.2 in [4], one can see that E+ U e3 u f u is a complete system of representatives for the block-orbits of a 3-pyramidal STS(v) under the action of G on F U G defined as in the proof of Lemma 1.1. □ Remark 2.2. Considering that "cyclic STS" means "0-pyramidal STS under the cyclic group", as a very special case of the above theorem we have the well known fact that any cyclic STS(6n+1) is equivalent to a (Z6n+1, {1}, 3,1)-DF and that any cyclic STS(6n+3) is equivalent to a (Z6n+3, {3}, 3,1)-DF. Example 2.3. The empty-set clearly is a (Zn, {22"-1}, 3,1)-DF since every non-zero element of Zn is an involution. It is not difficult to see that one of the associated (2n — 1)-pyramidal STS(2n+1 — 1) is the point-line design of the n-dimensional projective geometry over Z2. Let D2n be the dihedral group of order 2n, namely the group with defining relations D2n = (x, y | y2 = xn = 1; yx = x-1y). We give here an example of a STS(3f) which is f -pyramidal under D2 f. Example 2.4. Let f = 1 or 3 (mod 6) but f = 9. Let ^ : Zf —> D2f be the group monomorphism defined by ^(i) = xi for each i G Zf. The hypotesis on f guarantees, in view of Peltesohn's result, that there exists a cyclic STS(f). Thus, by Remark 2.2, there exists a (Zf, {1}, 3,1)-DF or a (Zf, {3}, 3,1)-DF F according to whether f = 1 or 3 (mod 6), respectively. It is then obvious that {^(B) | B G F} is a (D2f, {2f, 3e}, 3,1)-DF with e = 0 or 1, respectively. M. Buratti el al.: 3-pyramidal Steiner triple systems 99 Thus there exists an f -pyramidal STS(3f) under the dihedral group D2 f for any f = 1 or 3 (mod 6) but f = 9. If, in the above example, we put f = 3, we obtain a representation of the affine plane of order 3 as a 3-pyramidal STS(9) under D6. In this case the difference family F is empty because it is relative to a spread which is not partial; its subgroups {1, y}, {1, xy}, {1, x2y} and {1, x, x2} cover indeed all elements of D6. Following the instructions of the "only if part" of Theorem 2.1 the blocks of the STS(9) are: {toi, TO2, TO3}, {1, x, x2}, {y, xy, x2y}, {toi, 1, y}, {toi, x, x2y}, {TOi,x2,xy}, {^2,1, xy}, {to2, x, y}, {to2, x2, x2y}, {TO3,1, x2y}, {TO3, x, xy}, {TO3,x2,y}. In Section 4 we will make use of D6 again, for the construction of a 3-pyramidal STS(24n + 9) under ©6 x Z4„+i. 3 The "if part" In this section we determine the values of v for which a 3-pyramidal STS(v) cannot exist, we namely prove the "if part" of the main result Theorem 1.2. For this, we need two lemmas about elementary group theory. Lemma 3.1. If G is a group of order 24n + 18, then G has a subgroup of index 2. Proof. It is well known that a group of order twice an odd number has a subgroup of index 2. See, for example, [19, Exercise 262]. □ The next lemma makes use of the so-called "Burnside normal p-complement theorem" which is here recalled (see [19, Theorem 6.17]). Theorem 3.2. Let P be a Sylow p-subgroup of a finite group G. If CG (P) = NG (P), then P has a normal complement in G. Lemma 3.3. If G is a group of order 16n + 8 containing exactly 3 involutions, then G has a subgroup of index 2 containing exactly one involution. Proof. Let j 1, j2, and j3 be the three involutions of G and let H = (j 1, j2, j3} be the group they generate. We point out that H is a normal subgroup of G since it is generated by all the elements of order 2. Now, let P be a 2-Sylow subgroup of G. As P has order 8 and G contains exactly three involutions, by taking into account the classification of the groups of order 8, we have three possibilities: namely P is either isomorphic to the group Z4 x Z2 or P contains exactly one involution, i.e., it is P ~ Z8 or P ~ Q8 (the quaternion group of order 8). First of all we prove that it is necessarily P ~ Z4 x Z2 .By contradiction, assume that P contains exactly one involution. It is known that, in general, any two involutions of G generate a dihedral group; also, a dihedral group of order 2h contains at least h involutions. Therefore, either j j2} ~ D4 ~ Z2 x Z2 or j j2} ~ D6. In both cases, (j, j2} has three involutions, hence j3 G j j2} and H = j j2}. Since P contains just one involution, it is necessarily H = D6, otherwise P should contain a subgroup isomorphic to Z2 x Z2. 100 Ars Math. Contemp. 13 (2017) 107-123 Let T be the subgroup of H of order 3. As usual, we denote by NG(T) and CG(T) the normalizer and the centralizer of T in G, respectively. By the N/C-theorem, the quotient group NG(T)/Cg(T) is isomorphic to a subgroup of Aut(T) ~ Z2. Since T is the unique subgroup of H of order 3, T is characteristic in H and then it is normal in G, that is Ng(T) = G. Therefore, either CG(T) = G or CG(T) is a subgroup of G of index 2. In both cases, CG (T) is a normal subgroup of G of even order. The three involutions of G are pairwise conjugate, hence, they are contained in any normal subgroup of G of even order and then in CG(T). It then follows that T is central in H contradicting the fact that H is dihedral. We conclude that P ~ Z4 x Z2; in particular, P is abelian and contains a subgroup Q ~ Z4; also, P D H ~ Z2 x Z2 .To prove the assertion, we need to show that G has a normal subgroup O of order 2n +1. In fact, the semidirect product of O and Q will be a subgroup of G of index 2 containing exactly one involution. Recall that |G : CG (j )| is the size of cZ(jj), the conjugacy class of j in G, which cannot exceed the total number of involutions in G hence, |G : CG(ji)| < 3 for any i = 1,2,3. On the other hand, P is abelian hence, P < CG(jj) for any i = 1,2, 3. It then follows that either |G : CG(ji)| = 1 for any i = 1,2,3 or |G : CG(ji)| = 3 for any i = 1, 2, 3. We first deal with the former case in which all involutions of G are central. Since G/H has order 2d, d odd, then it has a subgroup of index 2 (see for example [19, Exercise 262]). In other words, there exists a normal subgroup N of G of index 2 which contains H. Since H is a central 2-Sylow subgroup of N, by Theorem 3.2, H has a normal complement O in N; in particular, O has order 2n +1. Now, suppose the existence of another subgroup K of N of order 2n + 1. Then K n O is normal in K and the quotient Qi = K/(K n O) should be isomorphic to Q2 = (K + O)/O. However, Q1 has odd order, while Q2 has even order. Therefore, O is the only subgroup of N of order 2n + 1, hence it is normal in G. We finally consider the case |G : CG(ji) | =3 for any i =1, 2, 3. By the N/C-theorem, G/Cg(H) is isomorphic to a subgroup of Aut(H) ~ Da. Since P < Cg(H) < CgU), we have that CG (H) = CG (j) for any i = 1, 2, 3. Now note that CG (H) satisfies the same assumption as G and all involutions are central in CG(H). Therefore, we can proceed as in the previous case to show that there is a normal subgroup Q of CG(H) of order . As before, it is not difficult to check that Q is the only subgroup of CG(H) of order , therefore it is normal in G. Set now_G = G/Qand P = Cg(H)/Q. Note that P ~ P, P is normal in G (i.e., Ng(P) = G) and G/P has order 3. Also, since P is abelian, then P < C^(P) hence, |G : Cq(P)| = 1 or 3. Considering that, by the N/C-theorem, G/Cq(P) is isomorphic to a subgroup of Aut(P), and that Aut(P) has order 8, we then have that |G : C^(P)| = 1. This means that P is central in G. Hence, G is the direct product of P by a normal subgroup O of order 3 which is the quotient by Q of a normal subgroup O of G of order 2n +1. □ Theorem 3.4. There is no 3-pyramidal STS(v) in each of the following cases: (i) v = 1 (mod 24); (ii) v = 13 (mod 24); (iii) v = 21 (mod 24); (iv) v = 27 (mod 48); M. Buratti el al.: 3-pyramidal Steiner triple systems 101 (v) v = 43 (mod 48). Proof. Cases (i)-(ii). Suppose that there exists a STS(v) with v = 24n +1 or v = 24n +13 which is 3-pyramidal under a group G. Therefore G is a group of order 24n - 2 or 24n +10 with exactly three subgroups of order 2. Now note that these subgroups are precisely the 2-Sylow subgroups of G since the order of G is divisible by 2 but not by 4. Hence n2 (G), the number of 2-Sylow subgroups of G, is 3. By the third Sylow theorem n2 (G) should be also a divisor of ^. We conclude that 3 should be a divisor of |G| which is clearly false. Case (iii). Now assume that there exists a 3-pyramidal STS(24n + 21). Then there exists a (G, {23, 3e}, 3,1)-DF F for a suitable group G of order 24n + 18 with exactly 3 involutions and a suitable e > 0. By Lemma 3.1 there is a subgroup S of G of index 2. Note that an element of G has odd or even order according to whether it is in S or not, respectively. Thus, in particular, the three involutions of G are all contained in G \ S while every subgroup of G of order 3 is contained in S. Thus the subset of G \ S which is covered by AF has size |G \ S| — 3 = 12n + 6. If B is any block of F having some differences in G \ S, then it necessarily has two points lying in distinct cosets of S in G. Thus, up to translations, we have B = {0, s,t} with s € S and t G G \ S. It follows that AB n (G \ S) = {±t, ±(s — t)}, hence AB has exactly four elements in G \ S. From the above two paragraphs we conclude that 12n + 6 should be divisible by 4 which is clearly absurd. Cases (iv)-(v). Assume that there exists a 3-pyramidal STS(v) with v = 27 or 43 (mod 48). Thus there exists a (G, {23, 3e}, 3,1)-DF F for a suitable group G of order 48n + 24 or 48n + 40 with exactly three involutions and where e > 0 in case (iv) or e = 0 in case (v). Note that in both cases we have |G| = 8 (mod 16) so that, by Lemma 3.3, G has a subgroup S of index 2 containing exactly one involution. Thus the subset of G \ S which is covered by AF has size |G \ S| — 2 = 24n + 10 or 24n +18. Then, reasoning as in case (iii), 24n + 10 or 24n + 18 should be divisible by 4 which is absurd. □ 4 The "only if part" For proving the "only if part" of our main theorem, we have to give a direct construction for a 3-pyramidal STS(v) whenever v is admissible and not forbidden by Theorem 3.4, hence for any v = 7, 9,15 (mod 24) and for any v = 3,19 (mod 48). The most laboured constructions are those for the last two cases where we will use extended Skolem sequences and extended Langford sequences. Given a pair (k, n) of positive integers with 1 < k < 2n + 1, a k-extended Skolem sequence of order n can be viewed as a sequence (si,..., sn) of n integers such that n U {si,si + i} = {1, 2,..., 2n +1}\{k}. i= i The existence question for extended Skolem sequences was completely settled by C. Baker in [1]. Theorem 4.1 ([1]). There exists a k-extended Skolem sequence of order n if and only if either k is odd and n = 0,1 (mod 4) or k is even and n = 2, 3 (mod 4). 102 Ars Math. Contemp. 13 (2017) 107-123 A k-extended Langford sequence of order n and defect d can be viewed as a sequence of n integers (¿1,..., ¿n) such that n U {¿i + i + d - 1} = {1, 2,..., 2n + 1}\{k}. i=1 We need the following partial result about extended Langford sequences by V. Linek and S. Mor. Theorem 4.2 ([11]). There exists a k-extended Langford sequence of order n and defect d for any triple (k, n, d) with n > 2d, n = 2 (mod 4) and k even. For general background concerning Skolem sequences, their variants, and their applications, we refer to [20]. We also refer to the recent survey [10] which, however, fails to mention some of our work; for instance, extended Skolem sequences have been crucial in our mentioned work on 1-rotational STSs [2, 4] and also in two papers [6, 22] dealing, more generally, with 1-rotational k-cycle systems. In the next lemma we combine Skolem sequences and Langford sequences to get the last ingredient that we need for proving the "only if part" of our main result. This lemma will be used, specifically, in the construction of a 3-pyramidal STS(v) with v = 3 (mod 96). Lemma 4.3. There exists a (Z12n, {3,4}, 3,1)-DF for any even n > 2. Proof. We have to construct a set Fn of 2n - 1 triples with elements in Z12n whose differences cover Z12n \ {0, 3n, 4n, 6n, 8n, 9n} exactly once. For the small cases n G {2,4,6,8,12,14,20} one can check that we can take Fn = {B1,..., B2n-1} with Bi = {0, i, bi} and the bis as in the following table: n (61,62, .. . ,&2n-l) 2 (5,9,13) 4 (40, 37, 34, 30, 38, 29, 28) 6 (13,16, 20,19, 26, 28, 30, 39, 34, 37,40) 8 (17, 20, 22, 25, 28, 33, 36, 34, 39,47,46, 52, 54,45, 53) 12 (66, 63, 37, 64, 38, 62, 39, 58,40, 59,41, 67,42, 68,43, 69,44, 70,45, 71,46, 57,47) 14 (76, 74,43, 70,44, 77,45, 73,46, 68, 47, 69,48, 78,49, 79, 50, 80, 51, 81, 52, 82, 53, 83, 54, 67, 55) 20 (110,103, 61,108, 62,102, 63,100, 64,105, 65,106, 66,107, 67, 98, 68, 99, 69,111, 70,112, 71,113, 72,114, 73,115, 74,116, 75,117, 76,118, 77,119, 78, 97, 79) For all the other values of n, set n = 2m and e = r + ( — 1)r where r is the remainder of the Euclidean division of m by 4. By applying Theorem 4.1 one can see that there exists a (2m + 1)-extended Skolem sequence (s1,..., sm+e) of order m + e. Also, by applying Theorem 4.2 one can see that there exists a (2m — 2e)-extended Langford sequence (¿1,..., ¿3m_e-1) of order 3m —e —1 and defect m + e +1. Then one can see that the desired difference family is the one whose blocks are the following: {0, —i, si + 4m — 1} with 1 < i < m + e; {0, —(m + e + i), ¿i + 6m + 2e} with 1 < i < 3m — e — 1. □ M. Buratti el al.: 3-pyramidal Steiner triple systems 103 Theorem 4.4. There exists a 3-pyramidal STS(v) for any admissible value of v not forbidden by Theorem 3.4. Proof. Considering that the admissible values of v are those congruent to 1 or 3 (mod 6), we have to prove that there exists a 3-pyramidal STS(v) in the following five cases: (i) v = 7 (mod 24); (ii) v = 15 (mod 24); (iii) v = 9 (mod 24); (iv) v = 3 (mod 48); (v) v = 19 (mod 48). Cases (i)-(ii). Assume that v = 7,15 (mod 24). Let F be an (H, {h}, 3,1)-DF with H = Z6n+1 and h =1 if v = 24n + 7, or H = Z3 x Z2n+1 and h = 3 if v = 24n + 15. In the former case the existence of F is guaranteed by Peltesohn's result (see Remark 2.2) while in the latter it is enough to take F = {{(0,0), (1, i), (1, —i)} | 1 < i < n}. The group G := Z2 x Z2 x H has order v — 3 and exactly 3 involutions. Thus, by Theorem 2.1, it is enough to exhibit a (G, {23, 3e}, 3,1)-DF for some suitable e. Such a DF is, for instance, the one whose blocks are: {(0,0)} x B with B gF; {(0,1,0), (1,0, h), (1,1, —h)} with h g H where H is a subset of H such that {{h, — h} | h g H} is the patterned starter of H, i.e., the set of all symmetric 2-subsets of H (see [8]). Case (iii). Assume that v = 9 (mod 24), say v = 24n + 9. The group G := D6 x Z4n+1 has order v — 3 and clearly has exactly three involutions. Thus, by Theorem 2.1, it is enough to exhibit a (G, {23,31}, 3,1)-DF (of course here the DF will "use" multiplication on the first component and addition on the second). For this, we have to distinguish three cases according to whether n is odd or congruent to 0 or 2 (mod 4). The three cases are very similar; the blocks of the desired difference family can be taken as indicated in the table below. Blocks n odd n = 0 (mod 4) n = 2 (mod 4) {(1, 0), (x,n), (x, 2n)} Yes Yes Yes {(y, 0), (1, - nf1), (1, ¿nf1)} Yes No No {(i, o), (x, - n), (x, ¿n)} No Yes Yes {(y, 0), (1,i), (1, 2n + 1 - i)} 1 < i < n i = n+i 1 < i < n 1 < i < n {(y, 0), (x,i), (x2, -i)} 1 < i < n 1 < i < n, i = n 1 < i < n, and i = 7n+2 {(y, 0), (x, -i), (x2,i)} n +1 < i < 2n n +1 < i < 2n and i = n n +1 < i < 2n and i = 7n+2 {(1, 0), (x,i), (x, 2n - i)} 1 < i < n - 1 1 < i < n - 1 and i = n 1 < i < n - 1 and i = n We note that the subgroup of order 3 which is not covered by the differences of the above families is, in any subcase, {(1,0), (x, 0), (x2,0)}. 104 Ars Math. Contemp. 13 (2017) 107-123 Case (iv). Assume that v = 3 (mod 48), say v = 48n + 3. The group G = Z4 x Z12n has order v - 3 and it has exactly three involutions. Then, by Theorem 2.1, it is enough to exhibit a (G, {23, 3}, 3,1)-DF. Subcase (iv.1): n is odd. Take a (2n + 1)-extended Skolem sequence (si,..., s2n-1) of order 2n - 1 (which exists by Theorem 4.1). Set n = 2t + 1 and check that the blocks of a (G, {23, 3}, 3,1)-DF are the following: {(0, 0), (1,0), (3, 6t + 3)}; {(0, 0), (1, 3t + 2), (1, —9t - 5)}; {(0, 0), (1,i), (3,12t + 7 — i)} {(0, 0), (1, 6t + 3 + i), (3, 6t + 3 — i)} {(0, 0), (0,i), (0, — si — 4t — 1)} with 1 < i < 6t + 3 and i = 3t + 2; with 1 < i < 6t + 2; with 1 < i < 4t +1. Subcase (iv.2): n is even. Take a (Z12n, {3,4}, 3,1)-DF F using Lemma 4.3. One can see that a (G, {23, 3}, 3,1)-DF is the one whose blocks are the following. {(0,0), (1,0), (1, 9n)}; {0} x B {(0,0), (1,i), (3, 6n +1 — i)} {(0,0), (1,i), (3, 6n — i)} with B € F; with 1 < i < 3n; with 3n +1 < i < 6n — 1. Case (v). Assume that v = 19 (mod 48), say v = 48n + 19. The group G = Z4 x Z12n+4 has order v — 3 and it has exactly three involutions. Then, by Theorem 2.1, it is enough to exhibit a (G, {23}, 3,1)-DF.Let (s1, s2,..., s2n) beany (n+1)-extended Skolem sequence of order 2n (which exists by Theorem 4.1). Then the required difference family is the one whose blocks are the following: {(0,0), (1,0), (1, 3n +1)}; {(0,0), (0,i), (0, — si — 2n)} {(0,0), (1,i), (3, 6n + 2 — i)} {(0,0), (1, 6n + 3 — i), (3, i)} with 1 < i < 2n; with 1 < i < 3n; with 1 < i < 3n + 1. □ 5 Conclusion Theorem 3.4 and Theorem 4.4 are the "if part" and the "only if part" of the main result Theorem 1.2 which therefore is now completely proved. The existence of a 3-pyramidal STS (v) can be summarized in the following table where, in the third column, we put a group acting 3-pyramidally on a STS(v). M. Buratti el al.: 3-pyramidal Steiner triple systems 105 v Existence Group 24n + 1 No - 24n + 3 Yes ^^ n is even Z4 x Z6n 24n + 7 Yes Z X Z6n+1 24n + 9 Yes D6 X Z4„+1 24n + 13 No - 24n + 15 Yes Z X Z3 X Z2n+1 24n + 19 Yes ^^ n is even Z4 X Z6n+4 24n + 21 No - Note that an abelian group of order 24n + 6 has only one involution so that there is no STS(24n + 9) which is 3-pyramidal under an abelian group. Thus we see, from the above table, that there exists a STS(v) which is 3-pyramidal under an abelian group if and only if v = 7,15 (mod 24) or v = 3,19 (mod 48). References [1] C. A. Baker, Extended Skolem sequences, J. Combin. Des. 3 (1995), 363-379, doi:10.1002/ jcd.3180030507. [2] S. Bonvicini, M. Buratti, G. Rinaldi and T. Traetta, Some progress on the existence of 1-rotational Steiner triple systems, Des. Codes Cryptogr. 62 (2012), 63-78, doi:10.1007/ s10623-011-9491-3. [3] M. Buratti, Recursive constructions for difference matrices and relative difference families, J. Combin. 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Doyen, A note on reverse Steiner triple systems, Discrete Math. 1 (1972), 315-319, doi: 10.1016/0012-365x(72)90038- 6. [10] N. Francetic and E. Mendelsohn, A survey of Skolem-type sequences and Rosa's use of them, Math. Slovaca 59 (2009), 39-76, doi:10.2478/s12175-008-0110-3. [11] V. Linek and S. Mor, On partitions of {1,..., 2m + 1} \ {k} into differences d,... ,d + m — 1: Extended Langford sequences of large defect, J. Combin. Des. 12 (2004), 421-442, doi:10.1002/jcd.20001. [12] G. J. Lovegrove, The automorphism groups of Steiner triple systems obtained by the Bose construction, J. Algebraic Combin. 18 (2003), 159-170, doi:10.1023/b:jaco.0000011935.37751. c5. 106 Ars Math. Contemp. 13 (2017) 107-123 [13] E. Mendelsohn, On the groups of automorphisms of Steiner triple and quadruple systems, J. Combin. Theory Ser. A 25 (1978), 97-104, doi:10.1016/0097-3165(78)90072-9. [14] M. Mishima, The spectrum of 1-rotational Steiner triple systems over a dicyclic group, Discrete Math. 308 (2008), 2617-2619, doi:10.1016/j.disc.2007.06.001. [15] R. Peltesohn, Eine Losung der beiden Heffterschen Differenzenprobleme, Compositio Math. 6 (1939), 251-257. [16] K. T. Phelps and A. Rosa, Steiner triple systems with rotational automorphisms, Discrete Math. 33 (1981), 57-66, doi:10.1016/0012-365x(81)90258-2. [17] A. Pott, A survey on relative difference sets, in: Groups, Difference Sets, and the Monster, De Gruyter, Berlin, volume 4 of Ohio State Univ. Math. Res. Inst. Publ., pp. 195-232, 1996, doi:10.1515/9783110893106.195. [18] A. Rosa, On reverse Steiner triple systems, Discrete Math. 2 (1972), 61-71, doi:10.1016/ 0012-365x(72)90061-1. [19] J. S. Rose, A Course on Group Theory, Cambridge University Press, Cambridge, 1978. [20] N. Shalaby, Skolem and Langford sequences, in: C. J. Colbourn and J. H. Dinitz (eds.), Handbook of Combinatorial Designs, Chapman & Hall/CRC, Boca Raton, Florida, Discrete Mathematics and its Applications, chapter 53, pp. 612-616, 2nd edition, 2007, doi: 10.1201/9781420049954. [21] L. Teirlinck, The existence of reverse Steiner triple systems, Discrete Math. 6 (1973), 301-302, doi:10.1016/0012-365x(73)90102-7. [22] S.-L. Wu and M. Buratti, A complete solution to the existence problem for 1-rotational k-cycle systems of Kv, J. Combin. Des. 17 (2009), 283-293, doi:10.1002/jcd.20217. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 107-123 On some generalization of the Möbius configuration Krzysztof Petelczyc Institute of Mathematics, University of Biatystok, Ciolkowskiego 1 M, 15-245 Biatystok, Poland Received 13 November 2014, accepted 28 January 2017, published online 19 February 2017 The Möbius (84) configuration is generalized in a purely combinatorial approach. We consider (2nn) configurations M(„,v) depending on a permutation p in the symmetric group Sn. Classes of non-isomorphic configurations of this type are determined. The parametric characterization of M(„,v) is given. The uniqueness of the decomposition of into two mutually inscribed n-simplices is discussed. The automorphisms of M(„,v) are characterized for n > 3. Keywords: Möbius configuration, (84 ) configurations, Möbius pair, n-simplex. Math. Subj. Class.: 51D20, 05B30, 51E30 1 Introduction The Mobius (84) configuration is a certain configuration in a projective 3-dimensional space consisting of two mutually inscribed and circumscribed tetrahedra (cf. [7]). Each vertex of one tetrahedron lies on a face plane of the other tetrahedron and vice versa. Configurations with parameters (n4) were studied in detail in [4], but this is not the case, since the Mobius (84) configuration is not a point-line structure. An important role of the theorem connected with the Mobius configuration (which says, roughly speaking, that the Mobius configuration "closes") in a projective 3-dimensional space was presented in [12]: it is equivalent to the commutativity of the ground division ring. In this paper we deal with two n-simplices (simplices with n vertices, n > 3)1 instead of two tetrahedra (4-simplices). The way how an n-simplex is inscribed into another we define by a permutation p in the group Sn. The generalization of the Mobius configuration we obtain, is a (2nn)-configuration and it will be referred to as a Mobius pair of n-simplices, E-mail address: kryzpet@math.uwb.edu.pl (Krzysztof Petelczyc) 1 In geometry an n-simplex usually means a simplex having n +1 vertices. Our definition is slightly different. Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 108 Ars Math. Contemp. 13 (2017) 107-123 or shortly a Möbius n-pair. Only a combinatorial scheme (an abstract incidence structure, see e.g. [2, 10]) of a Möbius n-pair is investigated and we do not discuss problems regarding embeddability into projective (or other) spaces. Although these problems have been partially solved in [5] (the case with p = id), they are interesting and still open in general. As we know from [6], in a projective space, up to an isomorphism there are five (84) point-plane configurations with the property that at most two planes share two points, and dually at most two points are shared by two planes. These are precisely those configurations with two mutually circumscribed tetrahedra, and thus all of them are sometimes called the Mobius configurations. It is also known (cf. [10]), that these (84) configurations correspond to conjugacy classes of the permutation group S4. We shall prove, that two Mobius n-pairs are isomorphic if and only if the permutations, that determine them, are conjugate. Another important impact of the permutation on the geometry of the Mobius n-pair is that the cycle structure of p is associated with circuits in the incidence graph of the Mobius n-pair. As we shall see, the decomposition of the points of the generalized Mobius configuration into two complementary and mutually inscribed simplices is, generally, a unique one. Exceptions appear "near" the classical case n = 4. Three of five (84) Mobius configurations contain at least two distinct pairs of complementary 4-simplices. The next problem, which is considered in the paper, involves Mobius subpairs of a Mobius n-pair. We simply delete some number of points and blocks of one n-simplex and the same number of points and blocks of the second n-simplex with a hope to obtain a Mobius pair again. The conditions, under which we get a subpair in the Mobius n-pair, are determined. In the last part we use most of the established properties to characterize the automorphism group of the Mobius n-pair for n > 3. 2 Definitions, parameters and basic properties By a configuration we mean any point-block structure M = (S, L}, where the blocks are subsets of the set of points, i.e. L C 2S. The rank of a point is the number of blocks containing this point, and dually the size of a block is the number of points contained in this block. Let n be a natural number and X be a set. The family of all n-subsets of the set X will be denoted by P„(X). Let n > 3. We say that a configuration M is an n-simplex iff |L| = n, there is a subset V G Pn(S) such that for every V' G Pn-i(V) there is a unique block L G L containing V', and the rank of each point s G S \ V is less than n - 1. Elements of V will be called vertices of the simplex, and blocks of the simplex are said to be its faces. We say that two configurations M1 = (S1, L1}, M2 = (S2, L2} are isomorphic (and we write M1 = M2) iff there exists a bijective map f: S1 —> S2 such that conditions k G L1 and f (k) G L2 are equivalent. In case M1 = M2 = M the map f will be called an automorphism of M. Let us consider two sets A = ja1,..., a„} and B = {b1,..., bn} suchthat A n B = 0. Let p G Sn be a permutation of the set I = {1,..., n}. Now we introduce the following sets: La := {A' U {bj : A' G P„-1(A) and a G A'}, Lb := {B' U {avW} : B' G P„-1(B) and b G B'}. The configuration M(„jV) := (A U B, la U lb }, K. Petelczyc: On some generalization of the Möbius configuration 109 will be called a Mobius n-pair. The Mobius configurations can be identified with the Mobius 4-pairs, which Levi graphs are Figures 1, 2, 3, 4, 5. All of them are also presented in [10]. In particular, M(4 id) is the classical (84) Mobius configuration. Let M be a Mobius n-pair. We write: A^ Bj for blocks of M not containing a.j, 6j, respectively; a-points, b-points, A-blocks, B-blocks for points in A, B, and blocks in LA, LB, respectively. The configuration M reflects the main abstract properties of the classical Mobius configuration. 1. The a-points yield a simplex in M: for any (n - 1)-subset A \ {a^ of the a-points there is a unique block of M, which contains this subset (Aj, a face of the simplex in question); the remaining points (b-points) yield another simplex. 2. The simplex with a-points and the simplex with b-points are mutually inscribed: on each face, Aj, of the first simplex there is a unique vertex (bj) of the second one; on each face, Bj, of the second simplex there is a unique vertex (av(j)) of the first simplex. Thus, we can decompose M into two complementary substructures SA(M) = (A, LA) and SB (M) = (B, LB), which we call simplices of M (although, formally, a block of each of them is not a subset of its points; there is one extra point on each of its faces). In the forthcoming part we will use the notion of the incidence graph (the Levi graph) Gm associated with M. Recall that a Levi graph is a bipartite graph with partition induced by points vs. blocks (cf. [9, 10]). Two of its vertices x, y are said to be adjacent (which is written x ~ y) if x is a point, y is a block (or vice versa) and x G y (or y G x). Otherwise x is not adjacent to y, which we write x ^ y. The rank of a vertex is the number of vertices adjacent to it. A vertex of Gm will be called point-vertex, block-vertex, a,b-vertex, A,B-vertex, or simply aj, bj, Aj, Bj as it corresponds to the point or to the block of M. The Levi graph associated with SA(M), SB (M) will be denoted by Gsa(m), Gsb(m), respectively. Remark 2.1. Let M be a Mobius n-pair. The Levi graph Gm has the following properties: (i) for X = A, B, every point-vertex from Gsx (m) is adjacent to all but one block-vertices from Gsx (m), and vice versa, (ii) for X, Y = A, B and X = Y, every point-vertex from Gsx (m) is adjacent to precisely one block-vertex from Gsy(m), and vice versa. Immediately from the definition of M(„iV), the number of its points coincides with the number of its blocks and equals 2n, and the rank of every point coincides with the size of every block and equals n. Thus the structures we investigate are (2nn)-configurations. A standard parametric question related to configurations is: what is the number of points that are contained in two distinct blocks, and dually: what is the number of blocks containing two distinct points. Proposition 2.2. Let k, l be two different blocks of the structure ffi(nif). Then |k n 1| G {0,1,2, n — 2}. If both k, l are A-blocks, or both k, l are B-blocks then |k n 1| = n — 2. Otherwise, k = Aj and l = Bj for some i, j G I, and the following equivalences hold (i) |Aj n Bj | =0 iff ^(j ) = i = j, (ii) |Aj n Bj | = 1 iff ^(j) = i = j or ^(j) = i = j, 110 Ars Math. Contemp. 13 (2017) 107-123 (iii) |A n Bj | = 2 iff fj ) = i = j. Proof. It is straightforward from the definition that if k, l are both A-blocks or B-blocks then k n l has n — 2 elements. Let k = Aj G and l = Bj G for some i, j G I. Let i = j. If f(j) = i then Aj n Bj = {b, avj)}. Otherwise, for f(j) = i, we get Aj n Bj = {6j}. Let i = j. If f (i) = i we obtain Aj n Bj = {av(j}. In case f (i) = i it holds Aj n Bj = 0. □ Each conjugacy class of Sn corresponds to exactly one decomposition of a permutation f G Sn into cycles, up to a permutation of the elements of I. Now we describe how the cycle structure of f is reflected in block paths of M(„,v). Fact 2.3. A permutation f contains a cycle of length k < n iff there is a closed path of length 2k consisting of blocks of M(„,v) such that every two consecutive blocks intersect in precisely one point of M(„,v). Proof. Assume that f contains the cycle (iii2... ik). Then ajj+1 G Aj. n Bj. and bjj+1 G Bj, Ajj+1 for each j < k. Thus, the closed path in question is the following: Aj1, Bj1, Aj2 , Bj2 , . . . , Ajfc , Bjfc . Now assume that there exists a closed path 11, l1,..., lk, l'k of blocks of M(n,v) such that every two consecutive blocks intersect in a point. By Proposition 2.2(ii) every two consecutive blocks of the path are Aj g La, Bj g with f (j) = i = j or f (j) = i = j. Suppose f (j) = i = j holds for the first two blocks of our path, namely l1 = Aj, l[ = Bj and f (i) = i for some i G I .To obtain |l'x n l21 = 1 we must have l2 = Aj with f (i) = j. Thus the next two blocks are l2 = Av(j), l2 = Bv(j) and f (f (i)) = f (i). In general we obtain j = Avj-1(j), lj = Bvj-1 (j) and f j-1(i) = fj-2(i) for every j = 2,..., k. To close the path we need fk(i) = i. Let us put i = i0. Then the cycle (i0, i1,..., ik-1), where ij = fj (i) for j = 0,..., k — 1, is one of the cycles in the cycle decomposition of f. □ As the configuration M(n v) is symmetric, it makes sense to consider the dual configuration Fact 2.4. The configuration Mj?n is isomorphic to M(„,v). Proof. It is easy to note that Mj?n = M(n,v-1). Consider a G Sn such that a(1) = 1 and a(m) = n — m + 2 for m G I\ {1}. Let x G {a, b, A, B}, i G I. Then F : xj ^ xa(j) is an isomorphism mapping M(n,v-1) onto M(„,v). □ The problem of two isomorphic Mobius n-pairs will be considered in general in the last section of the paper. Another parametric characterization is now a simple consequence of Proposition 2.2 and Fact 2.4. Proposition 2.5. Let x, y be two different points of M(„,v). There exist 0, 1, 2, or n — 2 blocks of M(„,v) containing x and y. 3 Hidden Mobius pairs The goal of this section is to characterize M = M(„,v) that can be transformed into Mobius pair with simplices distinct from SA(M), (M) by a decomposition of the points or by a deletion of some points and blocks. Informally, we say that these Mobius pairs are hidden in M. K. Petelczyc: On some generalization of the Möbius configuration 111 Figure 1: The Levi graph of M(4,id) (isomorphic to the hypercube graph Q4). Figure 2: The Levi graph of ffi(4,v) with p = (1234). Figure 3: The Levi graph of M(4jV) with p = (123)(4). Figure 4: The Levi graph of M(4,v) with p = (1)(2)(34). a\ ___a2 as ___04 bi 62 63 64 Ai A2 A3 A4 Bi B2 B3 B4 Figure 5: The Levi graph of M(4jV) with p = (12)(34). 112 Ars Math. Contemp. 13 (2017) 107-123 3.1 Möbius n-pairs with the special decompositions Let us start with the following combinatorial observation: Remark 3.1. The Möbius configuration M = M(4 id) can be presented in 3 distinct ways as two mutually circumscribed simplices such that each of them is distinct from Sa(M),Sb (M). One could say that there are four Mobius 4-pairs hidden in M(4,id). Let n > 4, M = M(„,v), and assume that it is possible to decompose the points of M into two complementary and mutually inscribed simplices Si(M), S2(M) suchthatSt(M) = SX(M) for each t = 1, 2, X = A,B. Such a decomposition will be called a special decomposition. Lemma 3.2. Let S1(M), S2(M) be two simplices, that arise from a special decomposition of M. (i) For each i G I, and each t =1,2, it is impossible to have both Bi,bi in St(M), or both Ai,ai in St(M). (ii) For each t = 1,2, the blocks of St(M) are two B-blocks and two A-blocks. Proof. The proof involves only S1(M), since the reasoning for S2 (M) will be the same. (i) Assume that S1(M) contains both of Bi,bi. Then also some aj is a point of S1(M) for j G I. Consider the graph Gm. The vertices Bi, bi are not adjacent, so from Remark 2.1(i) aj ~ Bi and j = ip(i). The unique block-vertex not adjacent to aj in Gsi(m) is Aj or Bs for some s = 4, there exists another block in Si(M), that is different from B^ Bs, Aj. We have two b-points in Si (M) so far, thus this block is a B-block. The B-vertex of Gm, that is associated with this block, must be adjacent to a^). So this block is Bj, which is already one of the blocks in S1(M) (comp. with the scheme presented in Figure 7), a contradiction. (ii) Let Bj be the unique B-block of S1 (M) for some i G I. Then the remaining blocks of S1(M) are A-blocks. In view of Lemma 3.2(i), there are n - 1 b-vertices in GSi(m): every A-vertex is associated with the b-vertex, which is not adjacent to it. For n > 4 a contradiction with Remark 2.1(i) arises: every b-vertex is adjacent to precisely one of A-vertices, and thus it is not adjacent to at least two A-vertices in GSi(m). Let S1(M) contain at least three B-blocks. Without loss of generality, assume B1, B2, B3 are blocks of S1 (M). From Lemma 3.2(i), b1, b2, b3 are not in S1(M). Thus, from Remark 2.1(i), S1(M) contains aj1, aj2, aj3 such that j = <(j) for j = 1,2,3. Every blockvertex Bj must be adjacent to at least two of the point-vertices ajj/ with j' = j. On the other hand, it is adjacent to at most one of them, what follows from Remark 2.1(ii) applied to Gm. This contradiction actually completes the proof as other cases run dually. □ By Lemma 3.2 we prove a generalization of Remark 3.1. Proposition 3.3. Let M = M(n,v). The following conditions are equivalent (i) there is a special decomposition of M, (ii) n = 4 and there is X c I such that |X | = 2 and <(X) = X. Proof. (i) ^ (ii): From Lemma 3.2(ii) we get n = 4, and two B-vertices and two A-vertices in GSi(m). Let (e.g.) B1,B2 be the B-vertices of GSi(m). In view of Remark 2.1(i), there are vertices x, y in GSi(m) such that x ~ B1, y ~ B2 and x ^ B2, y ^ B1. By Lemma 3.2(i), x = b2, y = b2, and thus x = aj, y = aj where <(1) = i, <(2) = j. Then two A-vertices in GSi(m) are As, At with s,t = i, j. The remaining two point-vertices must be of the form bs, bt> with s', t' = 1,2, since they must be adjacent to both of B1, B2. On the other hand, bs>, bt/ need to be adjacent to precisely one of As, At, so {s',t'} = {s,t}. Thus s,t = 1, 2, {1,2} = {i, j} = {<(1), <(2)}, and X = {1,2} is the required set. (ii) ^ (i): Assume, without loss of generality, X = {1, 2} and consider M = M(4,v) with <(X) = X. Take blocks B1, B2, A3, A4 and points av(1), av(2), b3, b4 of M, and consider Gm. We have B1 ^ av(2), B2 ^ av(1), and B1, B2 ~ b3, b4. Similarly A3 ^ b4, A4 ^ b3, and A3,A4 ~ av(1),av(2), since <(1),<(2) G {1, 2}. Thus the Levi graph we consider is a Levi graph of a 4-simplex. It is easy to verify that A1, A2, B3, B4 and b1, b2, a3, a4 form another 4-simplex. The two obtained simplices are mutually circumscribed. Indeed, B1, b2; B2, b1; A3, a4; A4, a3, and A1, av(1) (or A1, av(2)); A2, av(2) (or A2, av(1)); B3, b4; B4, b3 are all pairs of adjacent vertices representing blocks (points) of the first simplex and points (blocks) of the second simplex in each pair. In other words, we have found a special decomposition of M. □ Due to Proposition 3.3 there is a correspondence between the special decompositions of M(n,v) and 2-subsets of I preserved by <. The correspondence is established up to complements, since the special decompositions arise only for n = 4, and thus if < preserves a 2-subset of {1, 2,3,4} then it preserves its complement as well. So, directly from Proposition 3.3 we get: 114 Ars Math. Contemp. 13 (2017) 107-123 Corollary 3.4. All (up to an isomorphism) Möbius n-pairs with a special decomposition are the following: 1. M(4 ¡a) with 3 distinct special decompositions associated with X = {1, 2}, {1, 3}, {1, 4}, 2. M(4,(i3)(24)) with the special decomposition associated with X = {1,3}, 3. M(4,(i2)(3)(4)) with the special decomposition associated with X = {1,2}. 3.2 Subpairs of Möbius n-pairs Let M = M(„,v), n > 4, k > 3, k < n, and M' be a Möbius k-pair obtained from M by deleting 2(n - k) points and 2(n - k) blocks. We call M' a k-subpair of M. The blocks of M' are subblocks of M, that is every block of M' arises as a block of M with n - k points removed. The subblocks of the A-blocks, the B-blocks are called the A-subblocks, the B-subblocks, respectively. Let S1(M'), S2(M') be two simplices of M'. For any t =1, 2, X = A, B we write St(M') ^ Sx(M) if all the points and the blocks of St(M') are points and subblocks of SX(M). Otherwise we write Sj(M') ^ SX(M). For Y c I by ( \ Y we mean the restriction of ( to the set Y. In order to determine all Mobius n-pairs with k-subpairs we need to prove some auxiliary facts. Lemma 3.5. One of the following conditions holds (i) Si(M') ^ Sa(M) and S2(M') ^ Sb (M), (ii) S2(M') ^ Sa(M) and Si(M') ^ Sb (M), (iii) Si(M') ^ Sa(M),Sb (M) and S2(M') ^ SA(M),SB (M). Moreover, if M' satisfies (iii) then there is a special decomposition of M'. Proof. Let S1(M') ^ SA(M) and S2(M') ^ SB(M). So there is an a-point or A-subblock in S2(M'). We consider only the case with an a-point, as the case with an A-subblock is symmetric. From Remark 2.1(ii) applied to Gm, and Remark 2.1(i) applied to Gm', there are at most two B-subblocks in S2(M'). Since k > 3, there is at least one A-subblock in S2 (M'). Note that a unique A-subblock, which does not contain an a-point of S1(M'), is a block of S1(M'). Thus all the points of S1(M) are in an A-subblock of S2(M'). This yields a contradiction with Remark 2.1(ii). The proof for each of the remaining cases (i.e. S2(M') ^ Sa(M) and S1(M') ^ Sb(M), S1(M') ^ Sb(M) and S2(M') ^ SA(M),or S2(M') ^ Sb(M) and S1(M') ^ SA(M))is analogous. Let M' satisfy (iii). The steps of the proof of Lemma 3.2 can be repeated for simplices of M'. As a result we get k = 4, and two A-subblocks and two B-subblocks in each of simplices of M'. Let Y c I be the set of subscripts of A-subblocks and B-subblocks in one of these simplices. From the reasoning analogous to the first part of the proof of Proposition 3.3 we get that Y is the set of all the subscripts used for labelling the points and the blocks of M', and there is a two-element set X c Y such that ( \ Y (X) = X. Therefore, in view of Proposition 3.3, there is a special decomposition of M'. □ Lemma 3.6. If the number of deleted B-blocks and the number of deleted A-blocks coincide (and equals n — k), then there is X c I such that |X | = n — k and f(X) = X. K. Petelczyc: On some generalization of the Möbius configuration 115 Proof. Assume that Bil,..., Bin_k and Aj1,..., Ajn_k are removed blocks. Consider a vertex av(is ) with s = 1,... ,n — k of Gm' , and assume av(is ) is in GSl (M') (the case with a(¿1),... ,<(in-k)} = {¿1,..., in-k}, and X = {¿1,.. .,in-k} is the set from our claim. □ Let us present a condition, which is sufficient and necessary to find a k-subpair in M(n,V). Proposition 3.7. Let M = ffl(n,v). The following conditions are equivalent (i) there is M', which is a k-subpair of M, (ii) there is X c I such that |X| = n — k and <(X) = X. Furthermore, if (ii) holds then M' = M(k,v^ (AX)). Proof. (i) ^ (ii): By Lemma 3.5 M' satisfies one of (i) - (iii) of Lemma 3.5. In cases (i) and (ii) of Lemma 3.5 the numbers of A-blocks and B-blocks deleted from M coincide and are equal to n — k. The claim follows directly from Lemma 3.6. If case (iii) of Lemma 3.5 holds, then there is a special decomposition of M', and we get our claim by Proposition 3.3. (ii) ^ (i): Without any loss of generality, let X = {1,..., n — k}. Recall that the rank of every vertex in Gm is n. Observe the Levi graph obtained from Gm by removing the vertices ai, Ai and bi, Bi for every i G X, and all edges passing through these vertices. We denote this Levi graph by H. Note that av(i) is not a vertex of H, since <(i) G X. Let j G X and take Aj. Clearly Aj is a vertex of H. There are n — k edges joining Aj with all ai in Gm. Thus, the rank of Aj in H is n — (n — k) = k. Similarly we set ranks of the remaining vertices aj,bj, Bj of H. All these ranks are k. From this and the construction of H we get that H is the Levi graph of two mutually circumscribed k-simplices, where the way they are inscribed one into another is induced by the action of < on the set I \ X. Therefore H = Gm' for some M', which is a k-subpair of M. □ 4 Isomorphisms and automorphisms 4.1 Isomorphic Mobius n-pairs Recall that the Mobius (84) configurations (i.e. Mobius 4-pairs) correspond to conjugacy classes of the permutation group S4. In this section we generalize this property to all Mobius n-pairs. Let us start with a key lemma that gives an account on isomorphisms of configurations M(n,v) with the unique decomposition into two n-simplices. 116 Ars Math. Contemp. 13 (2017) 107-123 Lemma 4.1. Let f be an isomorphism mapping M(n,v) onto ffi(n,^). Assume that either n = 4 and both = id contain no cycle of length 2, or n > 5. There is a G Sn such that f (Bi) = Ba(i) for each i G I, or f (Bi) = A^) for each i G I. (i) If f (Bi) = Ba(i) then f (bi) = ba{i), f (Ai) = Aa{i), f (ai) = aa{i) for each i G I. (ii) If f (Bi) = Aa(i) then f (bi) = aa(i), f (Ai) = B^-i(a(i)), f (ai) = b^-i(a(i)) for each i G I. Furthermore, a<^> = ^a holds in both cases: (i) and (ii). Proof. Let M1 ■= ffl(n,tp) and M2 ■= Let i,j G I and Bi be an arbitrary B- block of Mi. Clearly, either f (Bi) = Bj for some B-block Bj of M2, or f (Bi) = Aj for some A-block Aj of M2. Assume that f (Bi) = Bj. In view of Corollary 3.4, both M1, M2 are Mobius n-pairs without the special decompositions. Thus all the B-blocks of M1 are mapped onto B-blocks of M2. We introduce a map a G Sn associated with f by the formula a : i ^ j iff f (Bi) = B ji for all i,j G I. Then f (Bi) = B^. Let us analyze graphs Gmi and Gm2 : f (bi) = ba(i) as bi, ba(i) are unique b-vertices not adjacent to Bi, Ba(i) respectively in graphs Gmi , Gm2; f (Ai) = Aa^i) as Ai, A^) are unique A-vertices adjacent to bi, ba(i) respectively in G mi, Gm2 ; f (ai) = aa(i) as ai, aa(i) are unique a-vertices not adjacent to Ai, Aa(i) respectively in Gmi, Gm2. °n the other hand, f (av(i)) = a^(a(i)) as av(i), a^(a(i)) are unique a-vertices adjacent to Bi, Ba(i) respectively in Gm1 , Gm2 . So aa(v(i)) = a^(a(i)) and thus a 4 and G Sn. The following conditions are equivalent: (i) = (ii) M(n,v) = ffl(nrf). Proof. Let M1 = M(n,v) and M2 = (i) ^ (ii): Let i G I, ai, bi be points and A^ Bi be blocks of M1. Consider a map f associated to the permutation a given by the formula f (xi) = xa(i) for x G {a, b}, K. Petelczyc: On some generalization of the Möbius configuration 117 which maps the points of M 1 onto the points of M2. Then f (Aj) = A^) and f (Bj) = Ba(i), as the conditions ai G Ai; b G Bi uniquely determine blocks Ai; Bi; respectively. Clearly, conditions bi G Ai and ba(i) € Aa(i) are equivalent. Note that aa(v(i)) G Ba(i) is equivalent to a^,^)) G Ba(i) as well, since ay = ^a. Thus f is the required isomorphism. (ii) ^ (i): We restrict ourselves to n > 5 since for n = 4 this fact is well known, as it was mentioned at the beginning of this section. Let f be an isomorphism mapping M 1 onto M2. By Lemma 4.1, there is a G Sn associated with f such that ay = ^a. □ According to Theorem 4.2, the number of non-isomorphic configurations M(n,v) is equal to the number of partitions p(n) of a positive integer n. There is the generating function, recursive formula, asymptotic formula, and direct formula for p(n) (cf. [1]). The increase of n implies quick growth of p(n): p(5) = 7,p(6) = 11,... ,p(100) = 190569292,... ,p(1000) = 24061467864032622473692149727991. 4.2 The automorphism group structure of a Mobius n-pair For n = 3 the structure M(n,v) consists of two mutually inscribed triangles. From [8] the automorphism group of M(3,v) is isomorphic to S3 x C2. From the original paper of Mobius [7] the automorphism group of M(4,id) has order 192. The Mobius configuration is also a particular case of the Cox configuration. Recall the definition of the Cox configuration (comp. [3]). Let X be a set with n elements. The incidence structure (Cx)x = (Cx)n = (U {^2fc+i(X) : 0 < k < n}, [P2k (X): 0 < k < n}, CUd) is the (2n-1n) configuration, which is called the Cox configuration. Since the automorphism group of (Cx)n is established in [11] and M(4,id) = (Cx)4 (see Figure 8), we get the following: Fact 4.3. The automorphism group of M(4,id) is isomorphic to S4 x C3. It follows from Theorem 4.2 that the centralizer of y in Sn consists of automorphisms of M(n,v) for any n. Nevertheless, we will give a detailed characterization of automorphism group of M(4,v) with y = id, and of M(n,v) with n > 5. Let M = M(n,v) and 1 < v1 < ... < vr be the lengths of the cycles which are contained in the cycle decomposition of y G Sn. Assume that there are mt cycles of length vt, so n = J2r=1 mtvt. In other words Vl Vl Vl V2 V2 V2 Vr Vr Vr y = y 11 y2• • • y;yi2y22 ••.ym22...yiry2r -y;,, where yk4 is a cycle of length vt for k < mt, t < r. In view of Theorem 4.2 we can assume, that each cycle consists of consecutive natural numbers. If we set Mk := J2i-1 mi vi + (k -1)vt + 1 then yk4: Mfc ^ Mfc + 1 ^ Mfc + 2 ^ ... ^ Mfc + (vt - 1) ^ Mfc, and the effective domain of yk4 is the set X^4 := {^k, Mk + 1, • • •, Mk + (vt - 1)} C I. Taking all the domains of all the cycles we obtain the family of pairwise disjoint sets XV1,..., X;1, XV2,..., X;22,..., X Vr,..., X;r that yields a covering of I. Thus for any cycle yk4 we have y£ (X^) = X^ and yk4 fA X= id. 118 Ars Math. Contemp. 13 (2017) 107-123 {1,3,4} {2,3,4} Figure 8: The Möbius configuration as (Cx)4. The points and the blocks of M can be identified with the sequences (t, k, i, e) such that t < r, k < mt, i = 0,..., vt — 1, and e G {1,2, —1, -2} according to the formula: (t, k, i, e) = Ai+ßi for e = 1, for e = —1, for e = 2, Bi+Mt for e = —2. (4.1) Let vt = K ,...ymt) G Cm, at G Smt, and v = (vl,...,vr) G Xrt=1Cm, a = (ai,..., ar) G X r=1Smt. With the pair (v, a) we associate the map f(va) as follows: f(v,a)((t,k,i,e)) = (t,at(k),i + v\ mod vt,e). In like manner we define the map g(v,a) by: )((t, k, i, e)) (t, at(k), i + vjt — 1 mod vt, —e) for e = 1, 2, (t, at(k), i + vt mod vt, —e) for e = —1, —2. (4.2) (4.3) Lemma 4.4. The map f(v,a) is an automorphism of M, which preserves each ofsimplices Sa, SB . Proof. It follows directly from (4.2), that f(v,a) maps SA onto SA, and f(v,a) maps SB onto SB. Let i G and j G I. Assume that bj G Bi. By (4.1), Bi = (t, k, i0, —2) for some i0 G {0,..., vt — 1}, and bj = (t', k', j0, —1) for some t' < r, k' < mt,, j0 G {0,..., vv — 1}. Then f (Bi) = (t, at(k),io + v^t( t) mod vt, —2) and f (bj) = (t', a^(k'),jo + v Set i' (io + v at( k) mod vt) + k) and j' ( jo + vt at, (k') mod vt', —2). t' (k') mod vf) + (k')>so K. Petelczyc: On some generalization of the Möbius configuration 119 f (Bi) = Bi, and f (bj) = Bj,. Recall that bj G Bi iff j = i. If j' = i' then: firstly t' = t, next at(k') = at(k) and thus k' = k, and finally jo = ¿0. It means that j = i, which yields a contradiction. Hence f (bj) G f (Bi). Let aj G Bi. Then j = p(i). We have av(i) = (t, k, i0 + 1 mod vt, 1), so f (av(i)) = (t, k, io + 1 + vk mod vt, 1) = a^,). Therefore f (avW) G f (Bi). The incidence (membership) relation is preserved by f(v,a) in case aj G Ai and in case bj g Ai as well, that can be easily proved by similar reasoning. □ Let vt = (v,..., v) for all t < r, and v = (vi,..., vr). Let us put g0 := g(o,id). t Lemma 4.5. The map g0 is an automorphism of M, which interchanges simplices SA, SB. Proof. Immediately from (4.2), g0 maps SA onto SB, and SB onto SA. We restrict our proof to the incidence relation involving the B-blocks, as the case with the A-blocks runs similarly. Let i G X^4. From (4.1) Bi is represented by the sequence (t, k, i0, -2) for some i0 G {0,..., vt - 1}. The points that belongs to Bi are bj with j G I \ {i} and av(i). Clearly, g0(bj) = aj G Ai = g0(Bi). We have av(i) = (t, k, i0 + 1 mod vt, 1) and thus 00(av(i)) = (t, k, i0, -1) = bi. Then finally g0(av(i)) G g0(Bi). □ Since g(v,a) = g0f(v,a), from Lemma 4.4 and Lemma 4.5 we infer that: Corollary 4.6. The map g(v,a) is an automorphism of M, which interchanges simplices Sa, SB . We write Mk4 for the set of all the points and the blocks of M labelled by the elements of the set , and MVt = {Mk4 : k < mt}. Lemma 4.7. Let f be an automorphism of M, which (1) maps the B-blocks onto the B-blocks, or (2) maps the B-blocks onto the A-blocks. There is v G X^C^4 and a G Xr=iSmt such that (i) f = f(v,a) in case (1), or (ii) f = g(v,a) in case (2). In particular, for each k < mt there is k' < mt such that f (Mk4) = Mk,. Proof. (i): Let i G X^4. Assume that f (Bi) = Bj for some j G I. According to (4.1) there is i0 G {0,..., vt - 1} such that Bi = (t, k, i0, -2), and j0 G {0,..., vt, - 1} such that Bj = (t', k', j0, -2) for some t' < r, k' < mt,. Then, by Lemma 4.1(ii) we get f ((t, k, i0,e)) = (t', k',j0,e) for each value of e. The unique B-block containing ai = (t, k, i0,1) is Bv-i(i) = (t, k, i0 - 1 mod vt, -2), and a unique B-block containing aj is Bv-ij) = (t', k, j0 - 1 mod vt,, -2). Hence, f maps (t, k, i0 - 1 mod vt, -2) onto (t', k', j0 - 1 mod vt,, -2), and f maps (t, k, i0 - 1 mod vt,e) onto (t',k', j0 -1 mod vt,, e) generally. By induction we get f: (t, k, i0 - u mod vt, e) ^ (t', k', j0 - u mod vt,, e) for all u = 0,..., vt - 1. 120 Ars Math. Contemp. 13 (2017) 107-123 This characterizes the action of f on M^, in particular, f (M^) C Mk''. Conversely, f-1 maps Bj onto Bj. By the reasoning, analogous to this, which has been already done, we come to f-1(Mk'') C M^. Consequently, f (Mk) = Mk'', and therefore t' = t since f is a bijection. It provides that f preserves the set M Vt. We define the map a e Smt associated with f f mv* by the formula a: k ^ k' iff f (Mk ) = Mk', for all k, k' < mt. Set v| = j0 — i0 mod vt. Finally the formula for f is the following: f: (t, k, i, e) ^ (t, a(k), i + v| mod vt, e) for all i = 0,..., vt — 1. (ii): Based on Lemma 4.5, g0f is an automorphism of M, which maps the B-blocks onto the B-blocks. Then, from Lemma 4.7(i), g0f = f(v,a) for some v e X^C^4 and a e X^=1 Smt, and thus f = g-1 f(v,a). Note that g-1 = g(i,id). Consequently, f = g(i,id)f(v,a) = gof(v+i,a) = g(v+i,a). What is more, f preserves the set MVt, that follows directly from (4.3). □ Now we characterize automorphisms of M(„,v), which can be uniquely decomposed into two mutually inscribed n-simplices. Theorem 4.8. Let M = M(n,v) and 1 < v1 < ... < vr be the lengths of the cycles in the cycle decomposition of f e Sn. Assume that either n = 4 and f = id contains no cycle of length 2, or n > 5. Then Aut(M) = 0^=1 x S„iz). Proof. Let F be an automorphism of M. By Proposition 3.3, there is no special decomposition of M. Thus, F either interchanges SA(M) with SB (M) or preserves each of them. According to Lemma 4.7 there is v0 e X^C^4 and a0 e Xr=1Smt such that F = f(vo,ao) or F = g(vo,ao) = g0f(vo,ao). Furthermore, every fKa), g(v,a) with v e X^C^-4 and a e Xr=1Smt is an automorphism of M by Lemma 4.4 and Corollary 4.6. Since, by Lemma 4.7, F preserves each of the sets MVt, we can restrict the proof to the one fixed set MVt. Thus, we assume that i = 0,..., vt — 1, k < mt. For the simplicity of the notation, we will write (a(k), i + va(k), e) instead of (t, at(k),i + v0,t(k) mod vt,e). Moreover, we identify f(v,a) with f(v,a) f M^t, and g(v,a) with g(v,a) f mv , so we assume v e C^4, a e Smt. Let w e C^4, ft e Smt and note that f(w,j3)f(v,a)((k, i, e)) = f(w,^)((a(k), i + vfc,e)) = (^a(k),i + vfc + wa(fc),e). Let 4>a : Smt —> Aut(Cm) be the map defined by ^a : (vi,..., vmt) ^ (v a(1), . . . , va (mt)). Then the formula for the composition of /(v,a) and /(w„s) is It is not difficult to check that go and /(v,a) commute. Note also that i /(-f ,id) if z is even, I g(- ti ,id) if z isodd. K. Petelczyc: On some generalization of the Möbius configuration 121 Let k' < mt'. We introduce the family of maps g0k ((k',i,e)) = ffo((k',i,e)) if k' = k, (k', i,e) otherwise. Then the following equalities hold {got : z = 0,..., 2vt - 1 and z is even} = {g§, : z = 0,..., 2vt - 1 and z is odd} = {/(v,id) : vfc' =0 for k' = k}, {g(v,id) : vfc' = 0 for k' = k}. Therefore, for each v g Cm we have /(v,id) = go1 go2 .. .gcC^ ' where all numbers zk = 0,..., 2vt - 1 are even. Likewise g(v,id) = gOl gO2 ... gem*, where all numbers zk are odd. Hence, for each F g Aut(M) there is v g C™4 and a g Smt such that F = /(v,id)/(o,a) or F = g(v,id)/(o,a). To complete the proof it suffices to determine the remaining compositions: /(v,id)/( /(0,a)/(0,ß) /(v,id)/(0,a) g(v,id)f(0,a) = g0/(v,id)/(0,a) /(v,id)g(w,id) = /(v,id)g0/( g(v, id)g(w,id) = g0/(v,id)g0/( y"(w+v,id), y"(0,ßa), /"(^(v^a^ g0/(0a (v),a) = g(0a(v),a^ g0/(v,id)/(w,id) = g0/(w+v,id) = g( w+v,id), g0/(w+v,id) = /(-1,id)/(w+v,id) □ The Mobius n-pairs, which automorphism groups are not characterized by Theorem 4.8, admit a special decomposition. We say that an automorphism f of a Mobius n-pair M yields a special decomposition of M if f maps the pair {SA, SB} onto a distinct pair of mutually inscribed simplices. Theorem 4.9. The automorphism group of M(4jV) is isomorphic to (i) (C4 © S2) x C2 if ^ G S4 contains precisely one cycle of length 2, (ii) (C2 x S2) x C2 if ^ G S4 contains two cycles of length 2. Proof. In view of Theorem 4.2, without loss of generality we can consider M i = M(4,V1) with y>i = (1)(2)(34) in case (i), and M2 = M(4jV2) with ^2 = (12)(34) in case'(ii) (comp. Figures 4, 5). Let Fs G Aut(Ms) for s = 1,2. By Corollary 3.4, there is the special decomposition of each of Ms. Thus, Fs maps the pair {SA, SB } onto {SA,SB} or Fs yields the special decomposition of Ms. In case Fs maps the pair {SA,SB} onto {SA, SB}, by Lemma 4.7, there is v0 G {0} x {0} x C2, a0 G S2 x {id} for M1, or vo G C2 x C2, ao G S2 for M2 such that Fs = f(vo,ao) or Fs = g(vo,ao) = gof(vo,ao), respectively for s = 1, 2. By Lemma 4.4 and Corollary 4.6 all maps Fsf(v,a), Fsg(v,a), where v G {0} x {0} x C2 and a G S2 x {id} if s = 1, or v G C2 x C2, a G S2 if s = 2, are automorphisms of Ms preserving the pair {SA, SB}. Based on the proof of Theorem 4.8, these maps form the group C4 © S2 if s = 1, and the group x S2 if s = 2. 122 Ars Math. Contemp. 13 (2017) 107-123 Consider the following two transformations: x ai a2 a3 a4 bi b2 b3 b4 Z(x) bi b2 a3 ai a2 b3 b4 f(x) a2 ai b3 b4 bi b2 a3 a4 The map f is an automorphism, which yields a special decomposition of Mi; and f is an automorphism, which yields a special decomposition of M2. Assume that Fs yields a special decomposition of Ms. Then F = /Fi and F2 = f F2', where Fs' is the automorphism of Ms given by (4.2) or (4.3). Let us set the commutativity rules in the automorphism group of Ms. By (4.1), the points of M1, M2 correspond to the sequences (t, i, k, e) with e = 1, -1. Using the convention introduced at the beginning of this paragraph we get t = 1,2, v1 = 1, v2 =2, m1 =2, m2 = 1 and X1 = {1}, X| = {2}, Xx2 = {3,4} for M1; t =1, v1 = 2, m1 = 2, and X2 = {1,2}, X2 = {3,4} for M2. To avoid any misunderstanding, in case M2 we will write Y2, Y22 instead of X2, X2 respectively. Then f maps the points of M1 by the formula: /((t,k,i,e)) = (t, k, i, —e) (t, k, i + 1 mod 2, e) (t, k, i, e) for i + G XJ^1, for e = 1, i + ^jj G X 2, for e = —1, i + ^jj G X 2. (4.4) The map f can be defined on points of M2 as: f ((t, k,i,e)) = ^ (t,k, i,e) (t, k, i, —e) (t, k, i +1 mod 2, e) for e = 1, i + G Y2, for e = —1, i + ^jj G Y2, for i + G Y22. (4.5) Note, that /2 = f2 = id. Hence the cyclic group generated by f and the cyclic group generated by f both coincide with C2. All the formulas for compositions of f with g0, and f with f = f(v,a) can be calculated using (4.4) and (4.5) (it is rather technical and thus omitted) and then we get ff = f/and Î90 = /g(0,id) = g(T(o,o,i)(0),id)L where T(0,0,! )(v) = T(0,0,! )((v i, vJ, v2)) = (vi, vJ, v2 + 1). Analogous calculation can be done for f. If we set T( 1, i)(v) = T( j ! )((v J, v2 )) = (v J + 1, v2 + 1), then fg0 = fg(0,id) = g(T(1, 1) (0),id)f, ff(v,a) = f(v,a) f if only a = id, ff(v,a) = g0f(v,a)f = g(v,a)f provided that i + ^jj. G Yj2 and a = (12), f f(v,a) = go 1 f(v,a)f = g(T(1,1)(v),a)f as long as i + ^jj. G Y,2 and a = (12). □ K. Petelczyc: On some generalization of the Möbius configuration 123 References [1] G. E. Andrews, The Theory of Partitions, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998, doi:10.1017/cbo9780511608650, reprint of the 1976 original. [2] A. Betten, G. Brinkmann and T. Pisanski, Counting symmetric configurations v3, Discrete Appl. Math. 99 (2000), 331-338, doi:10.1016/s0166-218x(99)00143-2. [3] H. S. M. Coxeter, Self-dual configurations and regular graphs, Bull. Amer. Math. Soc. 56 (1950), 413-455, doi:10.1090/S0002-9904-1950-09407-5. [4] B. Grünbaum, Configurations of Points and Lines, volume 103 of Graduate Studies in Mathematics, American Mathematical Society, Providence, Rhode Island, 2009, doi:10.1090/gsm/ 103. [5] H. Havlicek, B. Odehnal and M. Saniga, Möbius pairs of simplices and commuting Pauli operators, Math. Pannon. 21 (2010), 115-128, http://www.geometrie.tuwien.ac.at/ havlicek/pub/moebius.pdf. [6] D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie, Springer Verlag, Berlin, 1932, doi: 10.1007/978-3-662-36685-1. [7] A. F. Mobius, Kann von zwei dreiseitigen Pyramiden eine jede in Bezug auf die andere um-und eingeschrieben zugleich heißen?, J. Reine Angew. Math. 3 (1828), 273-278, doi:10.1515/ crll.1828.3.273. [8] K. Petelczyc, Series of inscribed n-gons and rank 3 configurations, Beitrage Algebra Geom. 46 (2005), 283-300, http://emis.ams.org/journals/BAG/vol.4 6/no-1/17. html. [9] T. Pisanski and M. Randic, Bridges between geometry and graph theory, in: C. A. Gorini (ed.), Geometry at Work, Math. Assoc. America, Washington, D.C., volume 53 of MAA Notes, pp. 174-194, 2000. [10] T. Pisanski and B. Servatius, Configurations from a Graphical Viewpoint, Birkhauser Advanced Texts Basler Lehrbucher, Birkhauser, Basel, 2013, doi:10.1007/978-0-8176-8364-1. [11] M. Prazmowska and K. Prazmowski, The Cox, Clifford, Mobius, Miquel and other related configurations and their generalizations, arXiv:1404.4353 [math.CO]. [12] K. Witczynski, Mobius' theorem and commutativity, J. Geom. 59 (1997), 182-183, doi:10. 1007/bf01229575. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 125-136 Domination game on paths and cycles Gasper Kosmrlj * AbeliumR&D d.o.o., Kajuhova ulica 90, Ljubljana, Slovenia Received 10 July 2015, accepted 8 February 2017, published online 22 February 2017 Domination game is a game on a simple graph played by two players, Dominator and Staller who are alternating in taking turns. In each turn a player chooses a vertex in such a way that at least one new vertex gets dominated by this move. The game ends when all vertices are dominated, and thus no legal move is possible. As the names of the players suggest, Dominator tries to finish the game as fast as possible, while Staller wants to prolong its end as long as she can. By Yg (y'g) we denote the total number of moves in the game when Dominator (resp. Staller) starts, and both players play according to their optimal strategies. In a manuscript from 2012, Kinnersley et al. determined Yg and Yg' for paths and cycles, but have not yet published this very important result. In this paper we give an alternative proof for these formulas. Our approach also explicitly describes optimal strategies for both players. Keywords: Domination game, game domination number, paths, cycles. Math. Subj. Class.: 05C57, 91A43, 05C69 1 Introduction The Domination game is a game played on a simple graph G by two players, Dominator and Staller. They are alternating in choosing vertices from G such that in every move at least one new, previously undominated, vertex gets dominated. We say that a vertex is dominated if it is either chosen or is a neighboor of a chosen vertex. The game ends when all vertices of G are dominated, that is when the set of chosen vertices forms a dominating set of G. As the names of the players suggest, Dominator wants to finish the game in the least possible number of moves, while Staller's goal is just the opposite: to play the game as long as she can. A game when Dominator makes the first move will be called a D-game, while S-game will denote a game when Staller starts. The game domination number jg (G) * The author acknowledges the project was financially supported by the Slovenian Research Agency under the grants P1-0297 and L7-5554. E-mail address: gasperk@abelium.eu (Gasper Kosmrlj) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 126 Ars Math. Contemp. 13 (2017) 107-123 (Staller-start game domination number y'b (G)) denotes the total number of moves made in the D-game (resp. S-game) on G when both players play optimally. Although the game was introduced not long ago by Bresar et al. in [4], it has already attracted several authors who have studied different aspects of the game, among which the most popular seems to be the so-called 3/5-conjecture posed by Kinnersley et al. in [14]. A major breakthrough was made by Bujtas in [5] by introducing a powerful greedy method for proving upper bounds of the game domination number. For other papers with results regarding the mentioned conjecture see [2,6,10]. Moreover, the greedy method has already been applied in [7] to improve upper bounds on the domination number of a graph. Other topics of the domination game that were studied include computational complexity of the game (see [1, 16]), metric properties with respect to the domination game (see [3]), the domination game played on disjoint union (see [9]), realizations of the game domination numbers (see [17]), and characterizations (see [15, 18]). Also, the game has motivated studies of new games such as the total version of domination game, introduced and studied in [11, 12], and the disjoint domination game introduced by Bujtas and Tuza in [8]. In this paper we will study the domination game on paths and cycles. Kinnersley et al. have in [13] already found formulas for the game domination number of these graphs. Since these results are fundamental for the theory of the domination game, it is rather unfortunate that they are not published yet. Moreover, their proof is analytical and does not offer us much of an insight into the optimal strategies of both players. The latter will implicitly follow from our proof in the next section. In the rest of this section we introduce some notation and results needed in the new proofs of formulas for paths and cycles. For a graph G and a subset of vertices S C V (G) we denote by G|S a partially dominated graph where the vertices of S are considered to be already dominated, and thus not need to be dominated in the course of the game. Note that S can be an arbitrary subset of V(G), and not only a union of closed neighborhoods of some vertices. From G|S we get the residual graph if we remove from G all edges between already dominated vertices, and all vertices v that cannot be chosen anymore, more precisely N [v] C S. Theorem 1.1 (Continuation Principle, [14]). Let G be a graph and A, B C V(G). If B C A, then 7s(G|A) < 7s(G|B) and ig(G|A) < ig (G|B). One of the corollaries of Continuation Principle is the fact that Yg and Yg never differ for more than one. Theorem 1.2 ([4, 14]). For any graph G, Y (G) - ig (G)| < 1. We say that a partially dominated graph G|S realizes a pair (k, £) G NxN if Yg (G|S) = k and y'9(G|S) = In the light of Theorem 1.2 we can now classify graphs into three different families, each of them realizing one of the possible pairs (k, k + 1), (k, k) and (k + 1, k). By PLUS we denote the family of all graphs that realize (k, k +1) for some positive k. Similarly we define EQUAL and MINUS. Lastly, we say that a graph is a no-minus graph if Yg(G|S) < ig(G|S) for every S C V(G). Theorem 1.3 ([14]). Forests are no-minus graphs. Some other families of no-minus graphs are given in [9], as well as the game domination numbers of the disjoint union of two no-minus graphs. If at least one of the components is an equal graph the following holds. G. Kosmrlj: Domination game on paths and cycles 127 Theorem 1.4 ([9]). Let G1|S1 and G2|S2 be partially dominated no-minus graphs. If G1|S1 realizes (k, k) and G2|S2 realizes (t, m) (where m e {t, t + 1}), then the disjoint union (G1 U G2)|(S1 U S2) realizes (k + t, k + m). In the second case, when both of the components are plus graphs we get the next result. Theorem 1.5 ([9]). Let G1|S1 and G2|S2 be partially dominated no-minus graphs such that G1 |S1 realizes (k, k + 1) and G2 |S2 realizes (t, t +1). Then k + t < ys((G1 U G2)|(S1 U S2)) < k + t +1, k + t +1 < Yg ((G1 U G2)|(S1 U S2)) < k + t +2. 2 Paths and cycles The following formulas for the game domination number of paths and cycles were proved in [13]: Yg (Pn) = Yg (Cn) Yg (Pn) Yg (Cn) In the course of the game on a path or on a cycle, we come across two special partially dominated paths. Let P^ denote a partially dominated path of order n+2 with both pendant vertices being already dominated, see Fig. 1. By Pn we denote a partially dominated path of order n + 1 where exactly one of the leaves is dominated, see Fig. 1 again. Note that in both cases n denotes the number of undominated vertices. ( \n 1 - 1; n = 3 (mod 4), | \n 1; otherwise, " n" 2 , f \n-11 - 1; n = 2 (mod 4), \ \n—11; otherwise. -o.................o-o-• •-a.................o-o Figure 1: Partially dominated paths P^ (left) and Pn (right) At first we prove the formulas for the game domination number of paths with both leaves dominated. Lemma 2.1. For every n > 0, we have \n 1 - 1; n = 3 (mod 4), Yg (Pn) \ \ § 1, otherwise, , (P„) = i \21 +1, n = 2 (mod 4), Yg (Pn) \ \n 1; otherwise. Moreover, for every i, j > 0 such that i + j = n, ir = (i mod 4) and jr = (j mod 4), we n n 128 Ars Math. Contemp. 13 (2017) 107-123 also have Yg (P'' U Pj'') = Yg (P" U Pj') Yg (P'') + Yg (Pj''); Yg (P'') + Yg (Pj'') + 1; Yg (P'') + Yg (pj''); Yg (P'') + Yg (Pj'') + 1; Yg (P'') + Yg (Pj'')+2; (ir ,jr) e {0,1} X {0,1, 2, 3} U {0,1, 2, 3} X {0,1}, (ir ,j) e {2, 3} X {2, 3}, (ir ,jr) e {0, 1} X {0, 1}, (ir jr) e {0, 1} X {2, 3} U {2, 3} X {0,1} U {(2, 2)}, (ir,jr) e {(2, 3), (3, 2), (3, 3)}. Proof. We prove all four formulas simultaneously by induction on the number of undomi-nated vertices n. It is easy to check by hand that all formulas hold for n < 8. Let us now assume that n > 9. Let us first consider a D-game on P,'. Let u be one of the (dominated) leaves, v its (unique) neighbor and w the neighbor of v different from u. Since N[u] C N[v] C N[w] U {u} holds, using Continuation Principle we get that Yg(P«'|(N[w] U {u})) < Yg(P''|N[v]) < Yg(P,"|N[u]), and thus we can assume that Dominator does not choose a leaf nor a leaf's neighbor in his first move on P,''. Hence we get that the residual graph after the first move is the disjoint union Pr'' U Ps'' where r, s > 0 and r + s = n — 3. More precisely, we get the following: Yg (Pn) = 1 + min{Yg (P" U Ps") | r + s = n — 3, r, s > 0} . -O- o- x -o- ■o P'' P'' Figure 2: Partially dominated path after Dominator choosing x on Pf We now consider four cases regarding the value of n mod 4. By induction hypothesis we get that the following holds for every k > 2. Yg (Pi'fc) = 1 + min {Yg (P^ U P^+i) | i + m + 1 = k, i, m > 0} U {Yg(P«+2 U P^m+3) | i + m + 2 = k, i, m > 0} = 1 + min {Yg(P«)+ Yg(PL+1) I i + m +1 = k, i,m > 0} U {Yg (P«+2) + Yg (Pim+3) + 2 | i + m + 2 = k, i, m > 0} 1 + min {2i + (2m + 1) | i + m + 1 = k, i, m > 0} U {(2i + 1) + (2m + 1) + 2) | i + m + 2 = k, i, m > 0} G. Kosmrlj: Domination game on paths and cycles 129 1 + min{2k - 1, 2k} = 2k, Yg (P4fc+l) = 1 + min {7g (P& u PL+2) I e + m +1 = k, e,m > 0} U {7g(P£+! u P^m+i) I e + m + 1 = k, e, m > 0} U {Yg(Pïi+3 U P^+s) I e + m + 2 = k, e, m > 0} 1 + min{2k, 2k, 2k} = 2k +1, Yg (P4fc+2) = 1 + min {Yg(P& U Pim+s) I e + m + 1 = k, e, m > 0} U {Yg(P£+i U Pim+2) I e + m + 1 = k, e, m > 0} 1 + min{2k, 2k + 1} = 2k + 1, Yg (P4fc+S) 1 + min {Yg(p£ UPim) I e + m = k, e,m > 0} U {Yg(P£+i U P^m+s) I e + m + 1 = k, e, m > 0} U {Yg(P«+2 U Pim+2) I e + m + 1 = k, e, m > 0} = 1 + min{2k, 2k + 1, 2k + 1} = 2k +1. Similarly we compute the game domination number of P' of the S-game. Since the moves available are the same as those in the D-game, the set of possible residual graphs after Staller's first move is the same as above, that is P''_ 1, P,"_2, P^s or Pr'' U Ps'', where r, s > 1 and r + s = n - 3. By Continuation Principle, Staller either plays on a dominated vertex (which is a leaf) or chooses such a vertex x that the residual graph of P^'|N [x] has two components, both of which are paths with both of their leaves already dominated. Continuation Principle assures us that choosing the neighbor of a leaf or the vertex on a distance two from a leaf is never better for Staller than choosing a leaf. Using induction hypothesis the following holds for every k > 2. yO (Pik ) 1 + max {Yg (Pi'(k-1)+s )}U {Yg(P£ U PL+i) I e + m + 1 = k, e, m > 0} U {Yg(P«+2 U PL+s) I e + m + 2 = k, e, m > 0} 1 + max {Yg (Pi(fc_i)+s )}U {Yg(Pii) + Yg(Pim+i) I e + m + 1 = k, e, m > 0} U {Yg (P&+2) + Yg (P-L+s) + 1 I e + m + 2 = k, e, m > 0} 130 Ars Math. Contemp. 13 (2017) 107-123 1 + max {2k - 1} U {2i + (2m +1) | i + m + 1 = k, i, m > 0} U {(2i + 1) + (2m + 1) + 1 | i + m + 2 = k, i, m > 0} 1 + max{2k - 1, 2k - 1, 2k - 1} = 2k, Yg (P'fc+i) = 1 + max {Yg (Pk )}U {Yg(P£ U Pim+2) | i + m +1 = k, i,m > 0} U {Yg(P«+i U Pm+i) | i + m + 1 = k, i, m > 0} U {Yg (P«+3 U Pi'm+s) I i + m + 2 = k, i, m > 0} 1 + max{2k, 2k - 1, 2k, 2k - 1} = 2k + 1, Yg (P4/C+2) = 1 + max {Yg(Pi'fc+i)}U {Yg(P£ U P^m+s) | i + m +1 = k, i, m > 0} U {Yg(P'i+i U Pf+2) I i + m + 1 = k, i, m > 0} = 1 + max{2k + 1, 2k - 1, 2k} = 2k + 2, Yg (Pi'fc+s) = 1 + max {Yg (Pi'fc+2)}U {Yg(P& U P") | i + m = k, i, m > 0} U {Yg(P«+i U P^'m+s) | i + m + 1 = k, i, m > 0} U {Yg(Pii+2 U Pim+2) | i + m + 1 = k, i, m > 0} = 1 + max{2k + 1, 2k, 2k, 2k + 1} = 2k + 2 . Next we show the formulas for the game domination number of the disjoint union P/' U Pj'. From the first two formulas proven above we can quickly deduce that P^' is an EQUAL if and only if k = 0 (mod 4) or k = 1 (mod 4). In the other two cases we get that Pk' is a PLUS. Since paths are no-minus graphs, by Theorem 1.4 it follows that formulas hold when at least one of the components is an EQUAL. It remains to prove the case when both components are PLUS. Let us assume first that i = 4i + 2 and j = 4m + 2, where i, m > 1. We would like to prove that Pzii+2 U Pz(m+2 realizes the pair (2i + 2m + 3,2i + 2m + 3). By Theorem 1.5 it follows that we only need to prove the lower bound for Yg and the upper bound for Yg' . For the former we need to prove that Staller can ensure that on P^i+2 U P,4m+2 at least 2m + 2i + 3 moves are played. Since we did not say anything about the relation between i and m we can without loss of generality assume that Dominator plays his first move in Pzii+2. Using Continuation Principle in the same way as above when proving the formula for Yg (P„), he has only two G. Kosmrlj: Domination game on paths and cycles 131 conceptually different options for his move so that the residual graph of -P^+2 after this move is either P^ U Pi's+3 or P^+i U Pi's+2, where r + s = i - 1 and r, s > 0. If Staller then chooses a dominated leaf in the component with the odd number of undominated vertices, by induction hypothesis and by Theorem 1.4 we get the following. Yg (P 4£+2 UP 4m+2 ) = 1 + min > 2 + min j(P4r U Pis+3 U P4m+2) U P4s+2 U P4m+2) Yg (p Yg (P^r U Pis+2 U Pim+2) Tg (P4r U P4s+2 U Pim+2) = 2 + (Yg (Pi; ) + Yg (P4S+2) + Yg (Pim+2) + 1) = 2+(2r + (2s + 1) + (2m + 1) + 1) = 2+(2i +2m +1) = 2i + 2m + 3 . Proof of the upper bound in the S-game follows similar lines. Staller has three conceptually different options for her first move, and then Dominator plays in the component with an odd number of undominated vertices such that he chooses a vertex at a distance 2 from a dominated leaf. By induction hypothesis and Theorem 1.4 we then get the following for every r, s > 0, r + s = i - 1 . Y /Yg (P&+1 U Pim+2) \ Yg (P«+2 U Pi'm+2) = 1 + max I Yg (P£ U P^+g U P^+2) I V Yg (Pi'r+1 U Pi's+2 U Pim+2)/ (Y'g (P4(£-1)+2 U P4m+2 ) \ Yg (Pi; U Pi's U P4m+2^ I Yg (P4(r-1) + 2 U P4s+2 U Pim+2) ) ( Yg (Pi(.-1)+2)+ Yg (Pim+2 ) + 1 \ = 2 + max I Yg (Pi'r) + Yg (P&) + Yg (Pim+2) + 1 I V Yg (Pi(;-1)+2) + Yg (Pi's+2) + Yg (Pim+2) + 2 J / (2(i - 1) + 1) + (2m +1) + 1 \ = 2 +max I 2r + 2s + (2m + 1) + 1 I V (2(r - 1) + 1) + (2s + 1) + (2m + 1) + 2 / = 2 + max{2i + 2m + 1, 2i + 2m, 2i + 2m + 1} = 2i + 2m + 3 . Finally, we prove that the disjoint unions Pi^+2 U Pim+3 and Pi^+3 U Pim+3 both realize the pair (2i + 2m + 3,2i + 2m + 4) for every i, m > 0. By Theorem 1.5 it is enough to prove the lower bound in the S-game presenting Staller's strategy. If she chooses a dominated leaf in Pim+3 in her first move, by the formulas proven above, we get that Yg(P«+2 U Pim+g) > 1 + Yg(P«+2 U Pim+2) = 1 + 2i + 2m + 3 = 2i + 2m + 4 and from here also Yg(Pi;+3 U Pim+g) > 1 + Yg(P«+g U Pim+2) = 1 + 2i + 2m + 3 = 2i + 2m + 4 . 132 Ars Math. Contemp. 13 (2017) 107-123 This concludes the proof. □ A direct corollary of the above lemma are formulas for the game domination number on cycles. Since cycles are vertex-transitive graphs any first move on a cycle leads to the same residual graph - a path with both leaves dominated. Theorem 2.2. For every n > 3, we have Yg (Cn) = Y'g (Cn) = Proof. By Lemma 2.1 it follows rn]- 1; n = 3 (mod 4), r n 1; otherwise, rn-T 1 - 1; n = 2 (mod 4), r ^ 1; otherwise. Yg (Cn) 1+Yg (pn-3) = 1 + r1 +1; n - 3 = 2 (mod 4), r1; -1- otherwise rn-11 +1; n = 1 (mod 4), r n-11; otherwise rn 1- 1; n = 3 (mod 4), r n 1; otherwise. where we get the last equality by listing the values of both functions for n = 4k, 4k + 1,4k + 2,4k + 3 and k > 0. Similarly, using Lemma 2.1 again we get the Staller-start game domination number. For every n > 3, we get that Yg (c„) = 1+Yg (pn- 3) 1+ r1- 1; n - 3 = 3 (mod 4), r1; otherwise rnf11 - 1; n = 2 (mod 4), rn-211; otherwise. □ The proof of Lemma 2.1 also gives us optimal strategies for both players. Choosing a vertex at a distance 2 from a dominated leaf is always optimal for Dominator, while playing on a dominated leaf in every move of the game is optimal for Staller. Since every residual graph of Pn that occurs during the game has at least one dominated leaf, both players can play in the same way as on Pn'. Hence the following lemma directly follows. Lemma 2.3. For every n, m > 0, we have Yg (Pn U Pm) = Yg (Pn' u Pm) = Yg (P^ U P^) and Yg (Pn u Pm) = Yg (Pn' u Pm) = Yg (Pn' u Pm). In particular, the last lemma says that Yg (Pn) = Yg (Pn) and Yg (Pn) = Yg (Pn) for every n > 0. Now everything is ready for the main theorem - the formulas for paths. G. Kosmrlj: Domination game on paths and cycles 133 Theorem 2.4. For every n > 0, we have Y g (Pn) l'g (Pn) \n]- 1; n = 3 (mod 4), \n 1; otherwise, 2 Proof. We can easily check by hand that the assertion holds for n < 4. Let us now assume that n > 5. In the first move Dominator can either dominate two vertices by choosing one of the leaves, or three vertices by choosing any other vertex. By using Continuation Principle we can assume that Dominator never chooses a leaf. More precisely, choosing the leaf's neighbor can never be worse for Dominator than choosing a leaf. Hence, in his first move exactly three vertices are dominated, and the residual graph after this move is the disjoint union Pr' U Ps' where r, s > 0 and r + s = n - 3. We consider four cases regarding the value of n mod 4. In every case we first use the above argument about Dominator's first move, and then apply Lemma 2.3 and Lemma 2.1 to get the result for every k > 1. Y g (Pifc) = 1+min Yg (P4fc+l) {Yg(P4* U P4m+i) | l + m +1 = k, l, m > 0} U {Yg(Pii+2 U P4m+3) I l + m + 2 = k, l, m > 0} 1 + min {Yg(P4'i+2 U P4m+3) I l + m + 2 = k, l, m > 0} {Yg (P4' U P4'm+i) | l + m + 1 = k, l, m > 0} U 1 + min {2l + 2m +1 | l + m + 1 = k, l, m > 0} U {2(l + m + 2) | l + m + 2 = k, l, m > 0} = 1 + min{2k - 1, 2k} = 2k, 1 + min {Yg(P4' U P4m+2) I l + m +1 = k, l, m > 0} U {Yg(P4'+i U P4m+i) | l + m + 1 = k, l, m > 0} U {Yg(P4'+3 U P4m+3) I l + m + 2 = k, l, m > 0} 1 + min{2k, 2k, 2k} = 2k +1, Yg (P4k+2) = 1+min {Yg(P4' U P4m+3) | l + m +1 = k, l, m > 0} U n {Yg(P4'+i U P4m+2) | l + m + 1 = k, l, m > 0} = 1 + min{2k, 2k + 1} = 2k + 1, 134 Ars Math. Contemp. 13 (2017) 107-123 Yg (P4fc+3) 1 + min {Yg(P4£ U P4m) I e + m = k, e,m > 0} U {y'(P4i+1 U P4m+3) I e + m + 1 = k, e, m > 0} U {y'(P4^+2 u P4m+2) I e + m +1 = k, e, m > 0} = 1 + min{2k, 2k + 1, 2k + 1} = 2k +1. We can see that Dominator's moves are similar to those in a game when one or two leafs are dominated. Since this is not the case in the S-game, we have to be more careful when considering moves of Staller on Pn. In the first move she clearly can not select a dominated leaf and dominate only one new vertex. Hence she either plays on a leaf and dominates two new vertices, or plays on a vertex of degree two, and thus splits the path into two (smaller) partially dominated paths. By using Lemma 2.3 and Lemma 2.1 we get the following for every k > 1 . Y' (P4k) 1 + max {Yg(P4(k-1)+2)} U {Yg (P* U P4m+1) I e + m + 1 = k, e, m > 0} U {Yg(P4*+2 u P4m+3) I e + m + 2 = k, e, m > 0} 1 + max {2k - 1}U {2e + 2m + 1 I e + m + 1 = k, e, m > 0} U {2(e + m + 1) + 1 I e + m + 2 = k, e, m > 0} 1 + max{2k - 1, 2k - 1, 2k - 1} = 2k, y' (P4k+i) = 1 + max {Yg(P4(k-1)+3)} U {Yg (P4* U P4m+2) I e + m + 1 = k, e, m > 0} U {Yg (P4*+1 U P4m+1) I e + m + 1 = k, e, m > 0} U {Yg(P4*+3 u P4m+3) I e + m + 2 = k, e, m > 0} 1 + max{2k - 1, 2k - 1, 2k, 2k - 1} = 2k + 1, y' (P4fc+2) = 1 + max {Yg (P4k )}U {Yg (P4* U P4m+3) I e + m + 1 = k, e, m > 0} U {Yg(P4m U P4m+2) I e + m + 1 = k, e, m > 0} 1 + max{2k, 2k - 1, 2k} = 2k + 1, G. Kosmrlj: Domination game on paths and cycles 135 Yg (Pfc+3) 1 + max {Yg (Pfc+i)}U |ys (P4* U PU | t + m = k, t, m > 0} U {Yg(P-4£+i U Pim+s) | t + m + 1 = k, t, m > 0} U {Yg(Pie+2 U Pim+2) | t + m + 1 = k, t, m > 0} 1 + max{2k + 1, 2k, 2k, 2k + 1} = 2k + 2 . □ The last proof does not only give us the game domination number of paths, but also tells us what are the optimal moves of both players. We can see that choosing one of the leaves is an optimal first move of Staller on Pn when n ^ 1 (mod 4). In the case when n = 1 (mod 4) holds, Staller's optimal move is to play on a vertex v such that the residual graph obtained from Pn |N [v] consists of exactly two components Pr' and Ps', where r+s = n-3, r = 1 (mod 4) and s = 1 (mod 4). References [1] B. Bresar, P. Dorbec, S. KlavZar and G. Kosmrlj, Complexity of the game domination problem, Theoret. Comput. Sci. 648 (2016), 1-7. [2] B. Bresar, S. KlavZar, G. Kosmrlj and D. F. Rall, Domination game: extremal families of graphs for the 3/5-conjectures, Discrete Appl. Math. 161 (2013), 1308-1316. [3] B. Bresar, S. KlavZar, G. Kosmrlj and D. F. Rall, Guarded subgraphs and the domination game, Discrete Math. Theor. Comput. Sci. 17 (2015), 161-168. [4] B. Bresar, S. KlavZar and D. F. Rall, Domination game and an imagination strategy, SIAM J. Discrete Math. 24 (2010), 979-991. [5] Cs. Bujtas, Domination game on forests, Discrete Math. 338 (2015) 2220-2228. [6] Cs. Bujtas, On the game domination number of graphs with given minimum degree, Electron. J. Combin. 22 (2015), Paper 3.29, 18 pp. [7] Cs. Bujtas and S. KlavZar, Improved upper bounds on the domination number of graphs with minimum degree at least five, Graphs Combin. 32 (2016), 511-519. [8] Cs. Bujtas and Z. Tuza, The disjoint domination game, Discrete Math. 339 (2016), 1985-1992. [9] P. Dorbec, G. Kosmrlj and G. Renault, The domination game played on unions of graphs, Discrete Math. 338 (2015), 71-79. [10] M. A. Henning and W. B. Kinnersley, Bounds on the game domination number, manuscript, 2014. [11] M. A. Henning, S. KlavZar and D. F. Rall, Total version of the domination game, Graphs Combin. 31 (2015), 1453-1462. [12] M. A. Henning, S. KlavZar and D. F. Rall, The 4/5 upper bound on the game total domination number, Combinatorica, to appear. [13] W. B. Kinnersley, D. B. West and R. Zamani, Game domination for grid-like graphs, manuscript, 2012. [14] W. B. Kinnersley, D. B. West and R. Zamani, Extremal problems for game domination number, SIAM J. Discrete Math. 27 (2013), 2090-2107. 136 Ars Math. Contemp. 13 (2017) 107-123 [15] S. KlavZar, G. Kosmrlj and S. Schmidt, On graphs with small game domination number, Appl. Anal. Discrete Math. 10 (2016), 30-45. [16] S. KlavZar, G. Kosmrlj and S. Schmidt, On the computational complexity of the domination game, Iran. J. Math. Sci. Inform. 10(2) (2015), 115-122. [17] G. Kosmrlj, Realizations of the game domination number, J. Comb. Optim. 28 (2014), 447-461. [18] M. J. Nadjafi-Arani, M. Siggers and H. Soltani, Characterisation of forests with trivial game domination numbers, J. Comb. Optim. 32 (2016), 800-811. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 137-165 Affine primitive symmetric graphs of diameter two* * Carmen Amarra t Institute of Mathematics, University of the Philippines Diliman C. P. Garcia Avenue, Diliman, Quezon City 1101, Philippines Michael Giudici, Cheryl E. Praeger Centre for the Mathematics ofSymmetry and Computation, The University ofWestern Australia 35 Stirling Highway, Perth, WA 6009, Australia Received 28 January 2016, accepted 15 May 2016, published online 22 February 2017 Let n be a positive integer, q be a prime power, and V be a vector space of dimension n over Fq. Let G := V x G0, where G0 is an irreducible subgroup of GL (V) which is maximal by inclusion with respect to being intransitive on the set of nonzero vectors. We are interested in the class of all diameter two graphs r that admit such a group G as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which G0 is a subgroup of either rL(n, q) or TSp(n, q) and is maximal in one of the Aschbacher classes Cj, where i G {2,4,5,6,7, 8}. We are able to determine all graphs r which arise from G0 < rL(n, q) with i G {2,4,8}, and from G0 < TSp(n, q) with i G {2,8}. For the remaining classes we give necessary conditions in order for r to have diameter two, and in some special subcases determine all G-symmetric diameter two graphs. Keywords: Symmetric graphs, Cayley graphs, quasiprimitive permutation groups, linear groups. Math Subj. Class.: 05C25, 20B15, 20B25 *This paper forms part of the first author's Ph.D., which is supported by an Endeavour International Postgraduate Research Scholarship (with UPAIS) and a Samaha Top-Up Scholarship from The University of Western Australia, and forms part of the Australian Research Council Discovery project DP0770915 held by the last two authors. t Corresponding author. E-mail address: mcamarra@math.upd.edu.ph (Carmen Amarra), michael.giudici@uwa.edu.au (Michael Giudici), cheryl.praeger@uwa.edu.au (Cheryl E. Praeger) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 138 Ars Math. Contemp. 13 (2017) 107-123 1 Introduction A symmetric graph is one which admits a subgroup of automorphisms that acts transitively on its arc set; if G is such a subgroup, we say in particular that the graph is G-symmetric. We are interested in the family of all symmetric graphs with diameter two, a family which contains all symmetric strongly regular graphs. We consider those G-symmetric diameter two graphs where G is a primitive group of affine type, and where the point stabiliser G0 is maximal in the general semilinear group or in the symplectic semisimilarity group. Our main result is Theorem 1.1. Those affine examples where G0 is not contained in either of these groups were studied in [2]. Theorem 1.1. Let V = F^ for some prime power q and positive integer n, and let G = V x G0, where G0 is an irreducible subgroup of the general semilinear group rL(n, q) or the symplectic semisimilarity group TSp(n, q), and G0 is maximal by inclusion with respect to being intransitive on the set of nonzero vectors in V. If r is a connected graph with diameter two which admits G as a symmetric group of automorphisms, then r is isomorphic to a Cayley graph Cay(V, S) for some orbit S of G0 satisfying (S) = V and S = -S, and one of the following holds: 1. (Go, S) are as in Tables 1.0.1 and 1.0.2; 2. G0 satisfies the conditions in Table 1.0.3; 3. G0 belongs to the class C9. Furthermore, all pairs (G0, S) in Tables 1.0.1 and 1.0.2 yield G-symmetric diameter two graphs Cay(V, S). Notation for Tables 1.0.1 and 1.0.2. The set Xs is as in (3.2) and Wg is as in (3.5) in Section 3.2, Ys is as in (3.7) in Section 3.3, c(v) is as in (3.9) in Section 3.4, S0 is as in (2.4) in Section 2.2, and S#, So and SK are as in (3.1) in Section 3.1. Cayley graphs are defined in Section 2.1. The graphs marked f did not appear in [2]. Table 1.0.1: Symmetric diameter two graphs from maximal subgroups of rL(n, q) Go n GL (n,q) S Conditions 1 GL (m, q) i Sym (t), mt = n Xs qm > 2 and s > t/2 2 GL (k, q) 1 min {fc, m} ts GL (n, q1/r) o Z?_i, r > 2 and n > 2 vGo as in (3.14) c(v) = r — 1 or c(v) = r t4 GL in, q1/r J o Z„_i, r = 2 or n = 2 vGo as in (3.14) c(v) = 1 5 (Zq_i o (Z4 o Qs)).Sp (2, 2), n = 2, q odd vGo v e V * 6 GL (m, q) i® Sym (2), m2 = n Ys s > m/2 t7 GU (n, q), n > 2 So, S# 8 GO (n, q), n = 3 and q = 3 So 9 GO (n, q), nq odd, n > 3 or q > 3 So, So, or SH 10 GO+ (n, q), n even, q odd, n > 2 or q > 2 So or S# 11 GO_ (n, q), n even, q odd, n > 2 So or S# C. Amarra et al.: Affine primitive symmetric graphs of diameter two 139 Table 1.0.2: Symmetric diameter two graphs from maximal subgroups of TSp(n, q) Go n GL (n, q) S Conditions 1 t2 ts 4 5 6 CTeAut(F, ) Go Sp (m, q)4 .[q - 1].Sym (t), mt = n Xs GL (m,q) .[2], 2m = n |J. (Zq—i o Qs).O- (2, 2), n = 2, q odd v GO+(n,q), n = 2 and q = 2 So GO+ (n, q), q and n even, n > 2 or q > 2 S0 or S# GO— (n, q), q and n even, n > 2 S0 or S# qm > 2 and s > t/2 Wß* qm > 2 and ß e Fq v e V * Table 1.0.3: Restrictions for remaining cases Go n GL (n,q) Conditions Restrictions 1 GSp (k, q) ®GO£(m,q), m odd, q> 3 GL (n,q1/r) o Z, q-1 S (Zq—i o R).Sp (2t,r), n = r4 4 (Zq—1 o R).Sp (2t, 2), n = r4 5 (Zq—1 o R).O- (2t, 2), n = r4 6 GL (m, q) i® Sym (t), m4 = n 7 GSp (m,q) i® Sym (t), m4 = n, q odd t > 3 Proposition 3.14 c(v) = r - 1, r Proposition 3.16 (2), (3), (4) R Type 1, t > 2 Proposition 3.23 (1) R Type 2, t > 2 Proposition 3.23 (2) R Type 4, t > 2 Proposition 3.23 (3) t3 Proposition 3.25 Proposition 3.26 2 The reduction to these cases is achieved as follows. It is shown in [1] that any symmetric diameter two graph has a normal quotient graph r which is G-symmetric for some group G and which satisfies one of the following: (I) the graph r has at least one nontrivial G-normal quotient, and all nontrivial G-normal quotients of r are complete graphs (that is, every pair of distinct vertices are adjacent); or (II) all G-normal quotients of r are trivial graphs (that is, consisting of a single vertex). The context of our investigation is the following. It was shown that those that satisfy (II) fall into eight types according to the action of G [7]. One of these types is known as HA (see Subsection 2.1). In this case, the vertex set is a finite-dimensional vector space V = Fd over a prime field Fp and G = V x G0, where V is identified with the group of translations on itself and G0 is an irreducible subgroup of GL (d,p) which is intransitive on the set of nonzero vectors of V. The irreducible subgroups of GL (d,p) can be divided into eight classes Cj, i e {2,..., 9}, most of which can be described as preserving certain geometric configurations on V, such as direct sums or tensor decompositions [3]. Note that, if a diameter two graph r is G-symmetric, then the stabiliser Gv of a vertex v is not transitive on the remaining vertices since Gv leaves invariant the sets of vertices at distance 1, and distance 2, from v. Thus, in our situation, the group G0 is intransitive on the set V#, where V# := V \ {0}, the set of nonzero vectors. In paper [2] we considered the graphs corresponding to the groups G0 which are maximal in their respective classes Cj, for i < 8, and which are intransitive on nonzero vectors. (We did not consider the last class 140 Ars Math. Contemp. 13 (2017) 107-123 C9 since the groups in this class do not have a uniform geometric description.) Several classes were not considered because the maximal groups in these classes are transitive on V#, namely, the maximal groups are (a) symplectic groups preserving a nondegenerate alternating bilinear form on V, and (b) "extension field groups" preserving a structure on V of an n-dimensional vector space over Fq, where qn = pd. The aim of this paper is to examine the cases not treated in [2], namely, G0 preserves either an alternating form or an extension field structure on V, and: (III) The group G0 is irreducible and is maximal in GL (d,p) with respect to being intransitive on nonzero vectors. All quasiprimitive groups of type HA are primitive; the condition of irreducibility of G0 is necessary to guarantee that G0 is maximal in G, and hence that G is primitive. In particular, since G0 is intransitive on V#, G0 does not contain SL (V) or Sp (V). The classification in [3] can be applied to the two groups rL(n, q) and GSp (d,p): the irreducible subgroups of rL(n, q) and of GSp (d,p) which do not contain SL (n, q) and Sp (d,p), respectively, are again organised into classes C2 to C9. Again we do not consider the C9-subgroups. Observe that of the maximal subgroups of rL(n, q) in classes C2 to C8, the only transitive ones are the C3-subgroups rL(m, qn/m) with n/m prime, and the C8-subgroup rSp(n, q) of symplectic semisimilarities. We avoid these possibilities by choosing q maximal such that qn = pd. We then consider the two cases: (1) where G0 < rL(n, q) and G0 does not preserve an alternating form on F^ and (2) where G0 < TSp(n, q). Note that in this case it is possible for d/n to be not prime, and it follows from the maximality of q that G0 is not contained in a proper C3-subgroup of rL(n, q) or TSp(n, q), respectively. Since G0 is irreducible and we are not considering C9-subgroups, we now have G0 a maximal intransitive subgroup in the C (for rL(n, q) or TSp(n, q)) for some i G {2,4, 5,6,7,8}. All such subgroups of rL(n, q) for which n = d and i = 5 are considered in [2]; moreover, for some of these cases, the arguments were given in the general setting of Cj-subgroups of rL(n, q), and so can be applied here. The cases requiring the most detailed arguments are those for subfield groups and, to a lesser extent, normalisers of symplectic-type r-groups (Crgroups with i G {5, 6}). As in [2], for each family of groups G0 we have two main tasks: (i) to determine the G0-orbits, and (ii) to identify which of these orbits correspond to diameter two Cayley graphs. In the instances where we are not able to achieve either of these, we obtain bounds on certain parameters to reduce the number of unresolved cases. The rest of this paper is organised as follows: In Section 2 we give the relevant background on affine quasiprimitive permutation groups, semilinear transformations and semi-similarities. In Subsection 2.3 we present Aschbacher's classification of the subgroups of rL(n, q) and TSp(n, q). Section 3 is devoted to the proof of Theorem 1.1, which we do by considering separately the maximal intransitive subgroups in each of the classes Cj, where i G {2,4, 5, 6, 7, 8}. Notation. If A is a vector space, a finite field, or a group, A# denotes the set of nonzero vectors, nonzero field elements, or non-identity group elements, respectively. The finite field of order q is denoted by Fq. The notation used for the classical groups, some of which is nonstandard, is presented in Section 2. If r is a graph, V(r) and E(r) are, respectively, its vertex set and edge set. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 141 2 Preliminaries 2.1 Cayley graphs and HA-type groups The action of a group G on a set Q is said to be quasiprimitive of type HA if G has a unique minimal normal subgroup N and N is elementary abelian and acts regularly on Q. The group G is then a subgroup of the holomorph N.Aut (N) of N (hence the abbreviation HA, for holomorph of an abelian group). It follows from [4, Lemma 16.3] that a graph r that admits G as a subgroup of automorphisms is isomorphic to a Cayley graph on N, that is, a graph with vertex set N and edge set {{x, y} | x — y G S} for some subset S of N# with S = -S and 0 G S. (Since N is abelian we use additive notation, and in particular denote the identity by 0 and call it zero.) Such a graph is denoted by Cay(N, S). If, in addition, r is G-symmetric, then S must be an orbit of the point stabiliser G0 of zero. Thus, in order for r to have diameter two, the group G0 must be intransitive on the set of nonzero elements in N. The result that is most relevant to our investigation is Lemma 2.1, which follows from the basic properties of Cayley graphs and quasiprimitive groups of type HA. Lemma 2.1 ([7]). Let r be a graph and G < Aut (r), where G acts quasiprimitively on V(r) and is of type HA. Then G = Fp x G0 < AGL (d,p) and r = Cay(F^S) for some finite field Fp, where the vector space F^ is identified with its translation group and G0 < GL (d,p) is irreducible. Moreover, r is G-symmetric with diameter 2 if and only if S is a G0 -orbit of nonzero vectors satisfying —S = S, S C V and S U (S + S) = V. The condition —S = S implies that |S + S| < |S|(|S| — 1) + 1,andif S is a Go-orbit then clearly |S| < |G0|. It follows from Lemma 2.1 that if Cay(V, S) is G-symmetric with diameter two then |V |<|S|2 + 1 <|Go|2 + 1. (2.1) This fact will be frequently used in obtaining bounds for certain parameters. In our situation pd = qn and G0 preserves on V the structure of an Fq-space; we therefore regard V as V = F^ and G0 as a subgroup of rL(n, q). 2.2 Semilinear transformations and semisimilarities Throughout this subsection assume that q is an arbitrary prime power, V is a vector space with finite dimension n over Fq, and B := {vi,..., vn} is a fixed Fq-basis of V. The general semilinear group rL(n, q) consists of all invertible maps h : V ^ V for which there exists a(h) G Fq, which depends only on h, satisfying (Am + v)h = Aa(h)uh + vh for all A G Fq and u, v G V. (2.2) The group rL(n, q) is isomorphic to a semidirect product GL (n, q) x Aut (Fq) with the following action on V: / n \9a n I £ AiVA := £ Afvf for all g G GL (n, q), a G Aut (Fq), and (2.3) \i=i J i= 1 Ai,...,An G Fq. If V is endowed with a left-linear or quadratic form then the elements of rL(n, q) that preserve ^ up to a nonzero scalar factor or an Fq-automorphism are called semisimilarities 142 Ars Math. Contemp. 13 (2017) 107-123 of That is, h is a semisimilarity of ^ if and only if for some A(h) G F# and some a'(h) G Aut (Fq), both of which depend only on h, ^(uh, vh) = A(h)^(u, v)a'(h) for all u, v G V if ^ is left-linear, and ^(vh) = A(h)^(v)a'(h) for all v G V if ^ is quadratic. It can be shown that a'(h) is the element a(h) in (2.2). The set of all semisimilarities of ^ is a subgroup of rL(n, q) and is denoted by H (n, q), where I is Sp, U, O, O+, or O-, if ^ is symplectic (i.e., nondegenerate alternating bilinear), unitary (i.e., nondegenerate conjugate-symmetric sesquilinear), quadratic in odd dimension, quadratic of plus type, or quadratic of minus type, respectively. The map a : rI (n, q) ^ Aut (Fq) defined by h ^ a(h) is a group homomorphism whose kernel GI(n, q) consists of all g G GL (n, q) that preserve ^ up to a nonzero scalar factor. The elements of GI(n, q) are called similarities of Likewise, the map g ^ A(g) for any g G rI (n, q) defines a homomorphism A from GI(n, q) to the multiplicative group F#. The kernel I( n, q) of A consists of all ^-preserving elements in GL (n, q), which are called the isometries of It should be emphasised that our notation for the similarity and isometry groups is non-standard, but follows for example [5]: the symbol GI(n, q) is sometimes used to denote the isometry group, whereas in the present paper this refers to the similarity group. In Subsection 3.1 we determine the orbits in V# of the groups rI (n, q). The following result, which gives the orbits of the isometry groups I(n, q), is useful: Theorem 2.2 ([8, Propositions 3.11, 5.12, 6.8 and 7.10]). Let V = and ^ a symplectic, unitary, or nondegenerate quadratic form on V. Then the orbits in V# of the isometry group of (V, are the sets S\ for each A G Im (^), where and SA := {v G V* | ¿(v) = A} (2.4) (¿(v,v) if $ is symplectic or unitary; (25) I ¿(v) if $ is quadratic. Observe that if $ is symplectic then ¿(v, v) = 0 for all nonzero vectors v, so it follows from Theorem 2.2 that Sp (n, q) is transitive on V*. 2.2.1 Some geometry Let f be a left-linear form on V. A nonzero vector v is called isotropic if f (v, v) = 0; otherwise, it is anisotropic. If f is symplectic or unitary, then an isotropic vector is also called singular. If f is symmetric bilinear and Q is a quadratic form which polarises to f (that is, f (u, v) = Q(u + v) - Q(u) - Q(v)), then a singular vector is a nonzero vector v with Q(v) = 0. Hence, in general, all isotropic vectors are singular and vice versa, unless V is orthogonal and q is even; in this case all nonzero vectors are isotropic but not all are singular. A subspace U of V is totally isotropic if f = 0, and totally singular if all its nonzero vectors are singular. On the other hand, a subspace U is anisotropic if all of its nonzero vectors are anisotropic. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 143 For any subspace U of V we define the subspace UL := {v € V | f (u, v) = 0 V u € U} and we write V = U ± W if V = U © W and W < U^. Clearly a nonzero vector v is isotropic if and only if v € (v)^, and the subspace U is totally isotropic if and only if U < UL. A symplectic or unitary form f, or a quadratic form with associated bilinear form f, is nondegenerate (or nonsingular) if the radical VL of f is the zero subspace. A hyperbolic pair in V is a pair {x, y} of singular vectors such that f (x, y) = 1. The space V can be decomposed into an orthogonal direct sum of an anisotropic subspace and subspaces spanned by hyperbolic pairs, as stated in the following fundamental result on the geometry of formed spaces. Theorem 2.3 ([6, Propositions 2.3.2, 2.4.1, 2.5.3]). Let V = F", and let f be a left-linear form on V which is symplectic, unitary, or a symmetric bilinear form associated with a nondegenerate quadratic form Q. Then V = (xi,yi)± ... ±( xm, ym) ^ U where {xj, yi} is a hyperbolic pair for each i and U is an anisotropic subspace. Moreover: 1. If f is symplectic then U = 0. Hence n is even and, up to equivalence, there is a unique symplectic geometry in dimension n over Fq. 2. If f is unitary then U = 0 if n is even and dim (U) = 1 if n is odd. Hence up to equivalence, there is a unique unitary geometry in dimension n over Fq. 3. If f is symmetric bilinear with quadratic form Q and n is odd, then q is odd, dim (U) = 1, and there are two isometry classes of quadratic forms in dimension n over Fq, one a non-square multiple of the other. Hence all orthogonal geometries in dimension n over Fq are similar. 4. If f is symmetric bilinear with quadratic form Q and n is even, then U = 0 or dim (U) = 2. For each n there are exactly two isometry classes of orthogonal geometries over Fq, which are distinguished by dim (U). In Theorem 2.3 (4), the quadratic form Q and the corresponding geometry is said to be of plus type if U = 0, and of minus type if dim (U) = 2. 2.2.2 Tensor products Some of the subgroups listed in Aschbacher's classification arise as tensor products of classical groups. In order to describe the group action we define first the tensor product of forms. If V = U ( W, and if and $w are both bilinear or both unitary forms on U and W, respectively, then the form ( $w on V is defined by (^u ( ^w) (u ( w, u' ( w') := (u, u')^w(w, w') for all u ( w and u' ( w' in a tensor product basis of V, extended bilinearly if and are bilinear, and sesquilinearly if and are sesquilinear. If and are both bilinear then so is ( ; moreover, ( is alternating if at least one of and 144 Ars Math. Contemp. 13 (2017) 107-123 Table 2.2.4: Tensor products of classical groups I(w,¿w ) I(U ( W,^u ( ^w) Sp Oe \ Sp if the characteristic is odd; |o+ else Sp Sp O+ | O+ if ej = + for some i, or e» = — for both i; Oei < O if dim (U) and dim (W) are odd; [O- else U U U is alternating, and ^u ( is symmetric if both and are symmetric. If ^u and are both unitary then ( is unitary. The tensor product I(U, ) ( I(W, ) acts on V with the usual tensor product action — that is, for any g e I(U, ), h e I(W, ), u e U and w e W, (u ( w)(g,h) := ug ( wh. The types of forms that arise according to the various possibilities for and , which are given in terms of the possible inclusions I(U, )(I(W, ) < I(V, ), are summarised in Table 2.2.4. The tensor product of an arbitrary number of formed spaces can be defined similarly: If V = Ui (•••( Ut and ^ is a nondegenerate form on U for each i, and either all ^ are bilinear or all are sesquilinear, the form ( • • • ( is given by t (®i=1&) (®t=iUi, ®t=iwi) = JJ ^(uj,Wj) i=1 as (t=1uj and (t=1wj vary over a tensor product basis of V, extended bilinearly if the ^ are bilinear, and sesquilinearly if they are sesquilinear. Then (t=1^i is a nondegenerate bilinear (respectively, sesquilinear) form on V. If the spaces (U», are all isometric, then we can extend the results of Table 2.2.4 to the following (see [6, 9]): t o / \ |Sp (mt, q) (t=1Sp (m,q) t \ IO+ (mt, q) {O (mt, q) O- (mt,q) O+ (mt,q) (t=1U (m, q) < U (mt, q) 2.3 Aschbacher's classification if qt odd; if qt is even if qm is odd; if e = - and t is odd; else The irreducible subgroups of semisimilarity and semilinear groups are classified by Aschbacher's Theorem [3]. In [6], Aschbacher's Theorem is used to identify those irreducible subgroups which are maximal. We present below the versions that correspond to rL(n, q) C. Amarra et al.: Affine primitive symmetric graphs of diameter two 145 and to rSp(n, q). Recall that G0 does not contain either of the transitive groups SL (n, q) or Sp (n, q). Theorem 2.4. If M is a maximal irreducible subgroup of rL(n, q) that does not contain SL (n, q), then M is one of the following groups: (C2) (GL (m, q) I Sym (t)) x Aut (Fq), where mt = n; (C3) rL(m, qr), where r is prime and mr = n; (C4) (GL (k, q) g GL (m, q)) x Aut (Fq), where km = n and k = m, and the action of t is defined with respect to a tensor product basis of F^ g F^; (C5) (GL (n,q1/r) o Zq-i) x Aut (Fq), where n > 2, q is an rth power and r is prime; (C6) ((Zq_i o R).T) x Aut (Fq), where n = rf' with r prime, q is the smallest power ofp such that q = 1 (mod r), and R and T are as given in Table 2.3.5 with R of type 1 or 2; (C7) (GL (m, q) I® Sym (t)) x Aut (Fq), where m1 = n, t > 2, and the action of t is defined with respect to a tensor product basis of g^F^; (Cg) TO (n, q) or rO±(n, q) with q odd, TSp(n, q), or rU (n, q); (C9) the preimage of an almost simple group H < PrL (n, q) satisfying the following conditions: (a) T < H < Aut (T) for some nonabelian simple group T (i.e., H is almost simple). (b) The preimage of T in GL (n, q) is absolutely irreducible and cannot be realised over a proper subfield of Fq. In Theorem 2.5 the symbol [o] denotes a group of order o. In case (C2) the group [q -1] is generated by the map 5M : Xi ^ ^Xi, yi ^ yi for all xi and all yi, i e {1,..., n/2}, where ^ is a generator of the multiplicative group F# and {xi,..., xn/2,yi,..., y„/2} is a basis of F£, satisfying ^(xi, Xj) = $(yi, yj) = ^(xi, yj) = 0 whenever i = j and ^(xi, yi) = 1 for all i. Such a basis is called a symplectic basis. Theorem 2.5. If M is a maximal irreducible subgroup of rSp(n, q), then M is one of the following groups: (C2) ((Sp (m,q)4 .[q - 1].Sym (t))) x Aut (Fq), where m = n/t; or (GL (m, q) .[2]) x Aut (Fq), where m = n/2; (C3) (Sp (m, qr) .[q — 1]) x Aut (Fq), where r is prime and m = n/r; or rU (m, q2), where m = n/2 and q is odd; (C4) (GSp (k, q) x GOe(m, q)) x Aut (Fq), where q is odd, k = m, m > 3, and GOe can be any of GO, GO+, or GO_; 146 Ars Math. Contemp. 13 (2017) 107-123 (C5) (GSp (n, q1/r) o x Aut (Fq) (C6) (Zq-1 o R) .O- (2t, 2), where q > 3 and is prime, and R is of type 4 in Table 2.3.5; (C7) (GSp (m, q) Sym (t)) x Aut (Fq), where qt is odd; (C8) rO±(n, q), where q is even; (Cg) the preimage of an almost simple group H < PrL (n, q) satisfying the following conditions: (a) T < H < Aut (T) for some nonabelian simple group T (i.e., H is almost simple). (b) The preimage of T in GL (n, q) is symplectic, absolutely irreducible, and cannot be realised over a proper subfield of Fq. Table 2.3.5: C6-subgroups R T Type 1 odd Ro o ■ ■ ■ o Ro, Ro := r++2 Sp (21, r) t Type 2 2 Z4 o Qs o ■ ■ ■ o Q8 Sp (21, 2) Type 4 2 D8 o ■ ■ ■ o D8 oQ8 O- (21, 2) r t1 3 Symmetric diameter two graphs from maximal subgroups of groups rL(n, q) and rSp(n, q) In this section we prove Theorem 1.1. In view of the observations in Section 1, assume that the following hypothesis holds: Hypothesis 3.1. Let V = Fp with p prime and d > 2, which is viewed as F^ with q = pd/n for some divisor n of d (possibly d/n composite or n = d). Let H be one of the subgroups below of GL ( d, p) : 1. H = rL(n, q) = GL (n, q) x (r}, the general semilinear group on V, or 2. H = rSp(n, q) = GSp (n, q) x (t}, the group of symplectic semisimilarities of a symplectic form on V, Let t denote the Frobenius automorphism of Fq and B be a fixed Fq-basis of V, with t acting on V as in (2.3) with respect to B (with g =1 and a = t); for the case where H = rSp(n, q) assume that B is a symplectic basis of V. Define G = V x G0 < V x H< AGL (d, p) and L = G0 n GL (n, q), where G0 is a maximal Cj-subgroup of H for some i e {2,4,5,6, 7, 8} and G0 does not contain Sp (n, q) or SL (n, q). C. Amarra et al.: Affine primitive symmetric graphs of diameter two 147 We note that the groups considered in [2] are the same as the subgroups L, as defined above, of H = rL(n, q). All irreducible subgroups of GL (d, p) which are maximal with respect to being intransitive on V# thus occur as subcases of the groups considered in Hypothesis 3.1 or belong to class C9. (Indeed, G0 is maximal intransitive if n = d or if d/n is prime.) For each Aschbacher class assume that G0 = M is of the form given in Theorem 2.4 or 2.5. Since some of the other subgroups of TSp(n, q) involve classical groups, we begin with class C8. 3.1 Class Cs In this case the space V has a form 0, which is symplectic, unitary, or nondegenerate quadratic if H = rL(n, q), and is nondegenerate quadratic if H = rSp(n, q) with q even. Since the symplectic group is transitive on V#, we consider only the unitary and orthogonal cases. Throughout this section we shall use the following notation: for 0 e {□, K, #} let Se := U Sa (3.1) AeF® where the Sa are as in (2.4). If q is a square (as in the unitary case), let q0 := Jq and let Fqo denote the subfield of Fq of index 2. Also let Tr : Fq ^ Fqo denote the trace map, that is, Tr(a) = a + aqo for all a e Fq. We begin by describing the orbits of the similarity groups GI(n, q), where I e {U, O, O+, O-}. Proposition 3.1. Let V = F^, 0 be a unitary or nondegenerate quadratic form on V, and G0 = GI(n, q) with I e {U, O, O+, O-}, according to the type of 0. Let S0 be as in (2.4) and Sa, S^ and S# be as in (3.1). 1. If 0 is unitary, then the G0-orbits in V # are S0 and S#. 2. If 0 is nondegenerate quadratic, then the Go-orbits in V # are as follows: (i) S# if n = 1; (ii) S0 and S# if n is even; (iii) S0, £□ and if n is odd and n > 3. Proof. Statement 2 is precisely [2, Proposition 3.9], so we only need to prove statement 1. Assume that 0 is unitary; hence q is a square and q0 = Jq. It follows from Theorem 2.2 that S0 is a G0-orbit (that is, provided that S0 = 0), so we only need to show that S# is a G0-orbit. Let v e S#; clearly, vGo Ç S#. For any u e S# set a := f (u, u)f (v, v)-1. Then a e F#, so a = ¡qo+1 for some 3 e Fq. Hence f (u,u) = 3qo+1f (v, v) = f (¡v,3v), so by Theorem 2.2 we have u = (¡v)9 for some g e U (n, q). Then u = v^9, where ¡g e GU (n, q). Therefore vGo = S#, which proves statement 1. □ The orbits of the semisimilarity groups can be easily deduced from Proposition 3.1. Proposition 3.2. Let V = F^, 0 be a unitary or nondegenerate quadratic form on V, and G0 = rI(n, q) with I e {U, O, O+, O-}, according to the type of 0. Then for all cases, the G0-orbits are exactly the same as the GI(n, q)-orbits. 148 Ars Math. Contemp. 13 (2017) 107-123 Proof. This follows from Proposition 3.1 and the fact that the elements of ri(n, q) preserve the form up to an automorphism of Fq. □ Hence, a direct consequence of Proposition 3.2 and [2, Proposition 3.12] is: Proposition 3.3. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with Go = rO (n, q) or G0 = rOe(n, q) (e = ±). Then r is G-symmetric with diameter 2 if and only if r = Cay(V, S) with V = F^ and the conditions listed in one of the lines 8-11 of Table 1.0.1 or lines 4-6 of Table 1.0.2 hold. We now consider the unitary case. Note that Theorem 2.3 implies that the space V contains a hyperbolic pair, which implies that there is some v e V which is nonsingular. The following are two easy but useful results which are analogous to Lemma 3.13 and Corollary 3.14 in [2]. Lemma 3.4. Let V = F^, ^ a unitary form on V, and ^ as in (2.5). Then Im = Fq0, the subfield of index 2 in Fq. Proof. Recall that f (v, v)^. = f (v, v) for any v e V, so Im < Fq0. By the preceding remarks V contains a nonsingular vector, say u. So f (au, au) = a^+1f (u, u) = n(a)f (u, u) for any a e Fq, where n : Fq ^ Fq0 is the norm map. Since n is surjective so is and the result follows. □ If ^(v, v) = 0, then (v}^ is nondegenerate and V = (v) ± (v)^. On the other hand, if ^(v, v) = 0 then (v) < (v}^. By the remarks in [6, pp. 17-18], the form ^ induces a nondegenerate unitary form on the space U := (v)^/(v), defined by (x + (v), y + (v)) := ^(x, y) for all x, y e (v)±. It follows from [6, Propositions 2.1.6 and 2.4.1] that all maximal totally isotropic subspaces of V have the same dimension, which, in all cases, is at most n/2, so in particular v^ contains a nonsingular vector whenever n > 3. Corollary 3.5. Let V = F^, ^ a unitary form on V, ^ as in (2.5), and v e V#. Then Im ) = Fq0 if v is nonsingular and n > 2, or if v is singular and n > 3. Proof. This follows immediately from Lemma 3.4 applied to (v)±, and the remarks above. □ Proposition 3.6. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with G0 = ru (n, q). Then r is G-symmetric with diameter 2 if and only if n > 2 and r = Cay(V, S), where V = F^ and S e {S0, S#}, with S0 and S# as in (2.4) and (3.1), respectively. Proof. By Lemma 2.1 and Proposition 3.1 we only need to prove that Cay(V, S) has diameter 2 if and only if n > 2. If n =1 then V is anisotropic, so GU (n, q) is transitive on V# by Proposition 3.1 (1) and Cay(V, S) is a complete graph. If n > 2 then V# \ S0 = S# and V# \ S# = S0 by Proposition 3.1. Claim 1: S# C S0 + S0. Let v e S#. Then by Corollary 3.5 there exists u e (v) with ^>(u) = —^>(v). Set w := ^(u + v), where ft := a^(v)-1 and a e Fq such that Tr(a) = ^(v). Then w, v — w e S0, so v e S0 + S0 and therefore S# C S0 + S0. Claim 2: S0 C SM + SMfor any ^ e (Im (^))#. Let v e S0. Suppose first that n > 3. _L Then by Corollary 3.5, for any ^ e (Im (^))# there exists w e SM n (v)±. It is easy to _ _ )X. verify that ^(v — w) = ^(w), so v — w e SM and v e SM + SM. Therefore S0 C SM + SM. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 149 If n = 2 then (v}^ = (v) for any v e S0. We show that there exists u e S0 such that ^(u, v) = 1. Indeed, take x e V \ (v). Then ^(v, x) = 0. If x e S0 define u' := x; if x G S0 let u' := av + ^(v, x)x where a e Fq with Tr(a) = —^(x). Then in both cases u' e S0 and ^(u', v) = 0, and we take u to be the suitable scalar multiple of u' such that ^(u,v) = 1. Let w := ^u + yv, where ^,7 e Fq with Tr(^) = 0 and Tr(^q°7) = Then w, v — w e SM, and thus v e SM + SM. Therefore S0 C SM + SM. It follows from Claims 1 and 2, respectively, that Cay(V, S0) and Cay(V, S#) both have diameter 2. This completes the proof. □ 3.2 Class C2 In this case V = ©f=1Uj, where Uj = F^ for each i, mt = n and t > 2. Assume that B = U t =1 Bj, where Bj is a basis for U for each i. We write the elements of V as t-tuples over F^; under this identification the t-action is equivalent to the natural componentwise action. Assume first that H = rL(n, q). It turns out that the G0-orbits in V# are the same as the L-orbits, and thus the graphs that we obtain are precisely those in [2, Proposition 3.2]. Lemma 3.7. Let G0 be as in case (C2) of Theorem 2.4. Then the G0-orbits in V # are the sets Xs for each s e {1,..., t}, where Xs := {(u1,..., ut) e V# | exactly s coordinates nonzero}. (3.2) Proof. Let v e Xs. Clearly vG° C Xs; since vL = Xs by [2, Lemma 3.1] it follows that vG° = Xs. □ Proposition 3.8. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1, with H = rL(n, q) and G0 as in case (C2) of Theorem 2.4. Then r is G-symmetric with diameter 2 if and only if r = Cay(V, Xs), where Xs is as in (3.2), such that qm > 2 and s > t/2. Proof. This follows immediately from Lemma 3.7 and [2, Proposition 3.2]. □ We now consider the case where H = TSp(n, q) with n > 4. By Theorem 2.5 there are two types of C2-subgroups, corresponding to two kinds of decompositions. We refer to these subcases as (C2.1) and (C2.2). (C2.1) The dimension m of the subspaces Uj is even, Uj is a symplectic space for each i, the subspaces Uj are pairwise orthogonal, and Go = {(gi,... ,gt)na | n e Sym (t), a e (t}, g e GSp (m, q), A(gj) = A(gi)} = (Sp (m,q)4 .[q — 1].Sym (t)) x (t}, (3.3) where A : GSp (n, q) ^ F# is as defined in Subsection 2.2. (C2.2) The dimension m = n/2 so that t = 2, both subspaces Uj are totally singular with dimension n/2, q is odd if n = 4, and Go = {(g,g-T) na | n e Sym (t), a e (t}, g e GL (m, q)} (34) = (GL (m,q) .[2]) x (t), (.) where gT denotes the transpose of g, and g-T = (gT)-1. 150 Ars Math. Contemp. 13 (2017) 107-123 Lemma 3.9. For each s € {1,..., t} let Xs be as in (3.2). The G0-orbits in V # are 1. the sets Xs for each s € {1,..., t} if case (C2.1) holds and G0 is as in (3.3); 2. the sets Xi and |J CTe/T\ Wp^ for all ft € Fq, if case (C2.2) holds and G0 is as in (3.4), where Wß := (wi,Xß)L (3.5) L = Go n GL (n, q) GL (m, q) .[2], wi := (1,0,..., 0) € Fm and xp € (F^)# with first component ft. Proof. The proof of part (1) is similar to that of [2, Lemma 3.1] and uses the transitivity of Sp (m, q) on U#, so we only need to prove part (2). Assume that case (C2.2) holds. Then L = K.Sym (2), where K := {(g, g-T) | g € GL (m, q)}. It is easy to see that Ui © {0} and {0} © U2 are K-orbits, so Xi = (Ui {0}) U ({0} © U2) is a Go-orbit. Let (u, v) € X2, and for any ft € Fq define wß := (ft, 0,..., 0) if ß = 0, (0,1, 0,..., 0) if ß = 0. (3.6) Since wi € uGL(m,q) we can assume that u = wi. Suppose that v = (ft, v2,..., vm). Claim 1: (wi, y) € (wi, v)K if and only if y = (ft, y2,..., ym) for some y2,..., ym € Fq. Indeed, (wi, y) € (wi, v)K if and only if y = vh for some h € StabGL(miq)(wi). Now wh = wi if and only if the matrix of h-T has the form 1 C \ 0 D 0 ) where C is a 1 x (m — 1) matrix over Fq and D G GL (m — 1, q). Clearly, the orbit of v under the subgroup { h-T | h G StabGL(m,q) (w1)} is the set of all nonzero vectors in F^ with first component ß. Therefore Claim 1 holds. Claim 2: (w1, v)L = (w1, v)K. By Claim 1 we can assume that v = wß. If ß = 0 let ß g := 0 0 Im— 1 If ß = 0 let g := (0 0) if m = 2, and 0 1 g := \ / if m > 2. Then g € GL (m, q) for all cases, and wg = wg = v. Hence (wg, vg ) = (v,wi), so that (v,wi) € (wi,v)K. Therefore (wi,v)L = (wi,v)K U (v,wi)K = (wi, v)K, which proves Claim 2. 0 0 0 Im-2 C. Amarra et al.: Affine primitive symmetric graphs of diameter two 151 It follows from Claims 1 and 2 that each set Wp is an L-orbit (and moreover Wp = Wp> if and only if p = p'). It follows from the definition of the t-action on V# that (wi,v)Go =U £ ■e Wp* . This completes the proof of part (2). □ Proposition 3.10. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with H = rSp(n, q) and i = 2. Then r is G-symmetric with diameter 2 if and only if r = Cay(V, S), where 1. if case (C2.1) holds, then qm > 2, G0 is as in (3.3), S = Xs, and s > t/2; 2. if case (C2.2) holds with qm = 2, then G0 is as in (3.4), and S = Wp for any ft G Fq ; 3. if case (C2.2) holds with qm > 2, then G0 is as in (3.4), and S = X1 or S = LU(r > Wp* for some ft G Fg ; with Xs as in (3.2) and Wp as (3.5). Proof. The graph of (1) is precisely that of Proposition 3.8, and the fact that it is G-symmetric follows from Lemma 2.1. So assume that case (C2.2) holds. By Lemma 2.1 we only need to show that V = S U (S + S) unless S = X1 and q = 2. It follows from Proposition 3.8 (with t = 2) that Cay(V, X1) has diameter 2 (with G quasiprimitive) if and only if qm > 2, which proves part of statement (3). Thus we may assume that S = Uae(j> Wp* for some ft G Fq. It remains to prove that V = S U (S + S). Let wp be as in (3.6) and 7 G Fq, with 7 = ft. Define go := 11 0 1 and h0 := 0 -1 -1 70 where y0 := 1 - ft 1y if ft = 0 and y0 := 0 if ft = 0. If m = 2 let g := g0 and h := h0; if m > 3 define g and h by g0 0 0 Im-2 , g : = and / h := h0 \ 0 1 'm-2 / Then g, h G GL (m, q) for all m > 2, and wg + wh = w. Recall that q is odd if m = 2 so we can take x g (F^ )# where wp if ft = 0; (0, -y/2) if ft = 0 and m = 2; (0,0,1,0,..., 0) if p = 0 and m > 3. Then for all cases y := xg + xh has first component 7. Hence, applying Lemma 3.9, we have W7 = (wi, y)L C Wp + Wp for any 7 = p. Since also {0} U Xi C Wp + Wp, it follows that V = Wp U (Wp + Wp). Therefore V = S U (S + S), which completes the proof of parts (2) and (3). □ 0 152 Ars Math. Contemp. 13 (2017) 107-123 3.3 Class C4 In this case V = U < W = F^ < F^ with k, m > 2, and B is a tensor product basis of V, that is, B = {wj < wj | 1 < i < k, 1 < j < m}, where Bu := {ui,..., wfc} and Bw := {wi,..., wm} are fixed bases of U and W, respectively. We choose t to fix each of the vectors wj < wj. Then for any simple vector w < w G V, we have (w < w)T = wT < wT, and for any v = J2¿=1 (a < G V, r vT = £ < < bT. j=i Recall that k = m in the description given in Theorems 2.4 and 2.5; however, all of the results in this section also hold for k = m, so we do not assume that k and m are distinct. In this way the results yield useful information for the C7 case. A nonzero vector in V is said to be simple in the decomposition U < W if it can be written as w < w for some w G U and w G W. The tensor weight wt(v) of v G V#, with respect to this decomposition, is the least number s such that v can be written as the sum of s simple vectors in U < W. It follows from [2, Lemma 3.3] that wt(v) < min {k, m} for any v G V#, and that for each s G {1,..., min {k, m}} there is a vector v G V# with weight s. The proof of the following observation is straightforward and is omitted. Lemma 3.11. For any v G V# and any a G Aut (Fq), wt(vCT ) = wt(v). Assume first that H = rL(n, q). As in the previous section, the G0-orbits in V# are the same as the L-orbits. This follows easily from Lemma 3.11 and the results in [2]. Lemma 3.12. Let G0 be as in case (C4) of Theorem 2.4. Then the G0-orbits in V# are the sets Ys for each s G {1,..., min {k, m}}, where Ys := {v G V#|wt(v) = s} . (3.7) Proof. This is a consequence of Lemma 3.11 above, and of [2, Lemmas 3.3 and 3.4]. □ We then obtain the same graphs as those in [2, Proposition 3.5]. Proposition 3.13. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with G0 as in case (C4) of Theorem 2.4, where k and m may be equal. Then r is G-symmetric with diameter 2 if and only if r = Cay(V, Ys), where s > ^ min {k, m} and Ys is as in (3.7). Proof. This follows immediately from Lemma 3.12 and [2, Propositon 3.5]. □ Now assume that H = rSp(n, q). In this case k is even, m > 3, q is odd, and ^ = ^u < &w, where is a symplectic form on U and is a nondegenerate symmetric bilinear form on W. We can choose Bu and Bw appropriately so that B is a symplectic basis and hence we can again choose t to fix each of the vectors wj < wj. The G0-orbits C. Amarra et al.: Affine primitive symmetric graphs of diameter two 153 in this case are proper subsets of the sets Ys in (3.7), and are in general rather difficult to describe, as are the L-orbits. For instance, if v = J2¿=1 a ® b € Ys, it is easy to see that vGo E ai ® bj aj € U #, bj € bf°i(m,q) „ i=1 If s = 1 then the set Y1 of simple vectors splits into the G0-orbits Y/, where 0 € {0, #} if m is even and 0 € {0, □, K} if m is odd, and Yl := {a b | a € U#, b € Se} . If s > 1 suppose that exactly r of the vectors bj belong in S# for some r, 0 < r < s; if m is odd suppose further that exactly rn belong in Sn and rH in SH. If m is even then vGo c Ysr, where Ysr := | ^ aj bj € Ys exactly r of the vectors bj are in S# | , and if m is even then vGo c YsrD'rH, where YsrD'r® := jE aj ® bj € Ys exactly re of the vectors bj are in Se for 0 € {□, K} j . The sets Ysr and Y/Dabove are, in general, not G0-orbits. For instance, if s = 2, the weight-2 vectors a1 ® b1 + a2 b2, al ® bl + a2 b2 € Y20 (or Y20'0 if m is even), such that b1 ± b2 and b1 / b2, belong to different G0-orbits. The following is an easy consequence of the preceding discussion. However, as discussed, we do not have a good description of the Go-orbits. Proposition 3.14. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with G0 as in case (C4) of Theorem 2.5, where k and m may be equal. If r is G-symmetric with diameter 2, then r = Cay(V, S) where S = vGo for some v € Ys, where Ys is as in (3.7) and s > 2min {k, m}. Proof. This follows immediately from the discussion above together with Proposition 3.13. □ 3.4 Class Cb In this case n > 2, d/n is composite with a prime divisor r, and V has a fixed ordered basis B := (v!,...,v„). Let q0 := q1/r and let Fqo denote the subfield of Fq of index r. Let V0 be the Fqo-span of B. Then V0 is a vector space over Fqo that is contained in V, but V0 is not an Fq-subspace of V. To any v = J2"=1 a^i € V we can associate the Fqo-subspace Dv of Fq, where Dv := (a1,...,a„)Fgo. (3.8) 154 Ars Math. Contemp. 13 (2017) 107-123 Set c(v) := dimFq0 (Dv ), (3.9) and note that c(v) < min {r, n}. For any A G Fq it is clear that DXv = ADv, so c(Av) = c(v), and it is also easy to show that c(vCT) = c(v) for any a G Aut (Fq). Let [Dv] := {ADv | A G F# }, and observe that Du g [Dv^ ] if and only if Du = ADv^ = 1 Dv j for some A G F#. Hence D (Duy Xa Dv, so that D -i G [Dv]. Thus [D^] = [Dv]a. 3.4.1 Case H = rL(n, q) By Theorem 2.4 Go = (GL (n,qo) o Z,_i) x (r> and L = GL (n, q0) o Zq-1. Regard the field Fq as a vector space of dimension r over Fq0, and for any a G {1,..., r}, define N Fq if a = r, K(a) = { q I Fqo otherwise. For a G {1,..., r} define where n(a) := |F# : K(a)#| (3.10) (3.11) i- 1 := qo - qô <=0^3 qô the number of a-dimensional subspaces of Fro. In particular n(r) = n(1) = 1. Lemma 3.15 gives some elementary observations about K(a) and n, whose significance will be apparent in Corollary 3.19. The proof of Lemma 3.15 is straightforward and is omitted. Lemma 3.15. Let Fq0 be a proper nontrivial subfield of Fq with prime index r, and suppose that Fq is viewed as a vector space over Fq0 with dimension r. For any a G {1,..., r}, let D denote the set of all Fq0 -subspaces of Fq with dimension a, and let K(a) and n(a) be as defined in (3.10) and (3.11), respectively. Then the following hold: 1. For any D gD, {X G Fq | XD = D} = K(a). 2. For any D G D, the sets [D] = {AD | A G F#} partition D. Moreover, |[D]| = |F# : K(a)# |, and the number of distinct parts [D] in D is n(a). The main result for this case, which relies on the value of the parameter c(v), is the following. It shows that examples do exist. r a C. Amarra et al.: Affine primitive symmetric graphs of diameter two 155 Proposition 3.16. Let r be a graph and G < Aut (T) such that G satisfies Hypothesis 3.1 with H = rL(n, q) and i = 5. Then r is connected and G-symmetric if and only if r = Cay(V, vGo ) for some v G V#. Moreover, if Dv and c(v) are as in (3.8) and (3.9), respectively, then the following hold. 1. If c(v) = r or c(v) = r — 1 then diam(r) = 2. 2. If c(v) = 1 then diam(r) = min {n, r}. In particular diam(r) = 2 if and only if n = 2 or r = 2. 3. If 2 < c(v) < 2min {n, r} then diam(r) > 2. 4. Let n be as defined in (3.11), s be the largest divisor of d/n with s < n(c(v)), and ,, , J 18s/17 if qo = 2; [s — 5/4 if qo > 2. If 3 < n < r and n/2 < c(v) < (r(n — 2) + ki(qo))/(2n), then diam(r) > 2. The cases not covered by Proposition 3.16 are discussed briefly at the end of the section. The proof of Proposition 3.16 is given after Lemma 3.20, and relies on several intermediate results. We begin by describing the GL (n, q0)-orbits in terms of the subspaces Dv, which in turn leads to a description of the G0-orbits in V#. Lemma 3.17. For any v G V# let Dv and c(v) be as in (3.8) and (3.9), respectively, and let U denote the set of all Fqo-independent c(v)-tuples in V0. Then for any fixed Fqo -basis {A,... ,Pc(v)} of Dv, ( c(v) vGL(n,qo) = Aui (ui,...,uc(v)) GU = {u G V# | D„ = Dv } . Proof. Suppose that v = ^"=1 a^. Define U := {u G V#|D„ = Dv } (3.12) and ( c(v) W := PiUi (ui,...,uc(v)) GU. (3.13) Claim 1: vGL(n,qo) Ç U. Let g G GL (n, q0) with matrix [gjk] with respect to B. Then vg = J2n=i a'kvk, where ak = ^"=1 ajgjk G Dv for each k. Hence DvS < Dv. Since v and g are arbitrary, we also have Dv < DvS. So DvS = Dv, and therefore vGL(n,qo) Ç U. Claim 2: U Ç W. Let u = J2"=i ajvj G U. Writing aj = J2C:SVi) P^j for each j, where all Yij G Fw, we get u = ^Él PiUi, with Ui = £j=i Yijvj G V0 for all i. It remains to show that the set u := {ui,..., uc(v)} is Fqo-independent. Indeed, let {ui,..., ub} be a maximal Fqo -independent subset of u, and extend this to an ordered Fqo -basis B' := (ui,..., ud) of V0. Then u = J2t=i Pku'k for some Pi,..., Pb G Fq, and 156 Ars Math. Contemp. 13 (2017) 107-123 if g € GL (n, q0) is the change of basis matrix from B' to B, then ug = ^fc=1 P'kvfc- So Du = Dug by Claim 1, and thus b < c(v) = dimF (Du) = dimF (Dug) < b. Hence b = c(v) and u is Fq0 -independent. Therefore U C W. Claim 3: W C vGL(n'qo). It is easy to see that W is contained in one orbit of GL (n, q0), and it follows from Claims 1 and 2 that v G W .So W C vGL(n'qo), as claimed. Thus we have vGL(n'qo) = U = W by Claims 1-3. □ Proposition 3.18. For any v € V# let Dv and c(v) be as in (3.8) and (3.9), respectively, and let U be the set of a// Fqo-independent c(v)-tuples in V0. Then for any fixed Fqo -basis i^i,... ,&(v)} of Dv we have c(v) A E ßi i=i (ui,...,uc(v)} eM, A G F* = {u G V* | Du = ADv, A G F*} and (ui, ...,uc(v) ) G M, A G F*, a G (t ) c(v) A E ß7 i=i {u G V* I Du = A(Dv)7, A G F*,a G (t)} . (3.14) Proof. Let U' := {u € V# | Du = ADv for some A € F#}. Since L = GL (n, qo) ◦ Z9_i and D^v = ADv for any A € Fq, it follows from Lemma 3.17 that vL = U'. Thus vGo = U {u7 7£{r > u G vL} Ç W' where W' := {u € V# | Du = A(Dv)CT, A € F#, a € (t)}. For any w € W with Dw = ^(Dv)p for ^ € F# and p € (t), we have w € (vp)L C vGo. Therefore vGo = W', and the rest follows from Lemma 3.17. □ Corollary 3.19. Let v € V#, and let K, n, Dv and c(v) be as defined in (3.10), (3.11), (3.8) and (3.9), respectively. 1. For a € {1,..., min {n, r}}, the number of orbits vL with c(v) = a is n(a). 2. |vL c(v) |GL (c(v),qo)| • |F* : K(c(v))*| 3. |vG s |vL | for some divisor s of d/n with s < n(c(v)). Proof. It follows from Proposition 3.18 that the map ^ [Dv] := {AD | A G F#} is a one-to-one correspondence between the set of L-orbits and the set of classes [D] of Fq0-subspaces of Fq. Therefore, by Lemma 3.15 (2), there are exactly n(a) orbits with c(v) = a, which proves part (1). Also by Proposition 3.18, we have |vL| = |U| • |[Dv]|, where U is the set of Fq0 -independent c(v)-tuples in V0. So |U| n c(v) |GL (c(v),qo) |, L v G o v= n C. Amarra et al.: Affine primitive symmetric graphs of diameter two 157 and by Lemma 3.155 (2), |[D„]| = |F# : K(c(v))#|. This proves part (2). Since L < Go we must have |vGo | = s |vL| for some s dividing |G0 : L| = |Aut (Fq) | = d/n. Also s < n(c(v)) since c(vCT) = c(v), which proves part (3). □ Lemma 3.20. Let r = Cay(V, vGo) for some v e V#, and let c(v) be as in (3.9). Let w e V. 1. If w e vGo + vGo then c(w) < 2c(v). 2. If Dw < Dv then w e vGo + vGo. Proof. Let U and W denote the sets of Fqo -independent c(v) - and c(w)-tuples, respectively, in V. o Suppose first that w = x + y for some x, y e vGo. Then by Proposition 3.18 we can write x and y as x = J2¿Si A/j^ and y = ^¿Si y/j7'yj for some scalars A, y e F#, maps p, a e Aut (Fq), and c(v)-tuples (x1,..., xc(v)) , (y1,..., yc(v)) e U. Hence Dw = Dx + y C (a/P, ..., A/P(v), u/i,..., u/%)) , and therefore c(w) = c(x + y) < 2c(v). This proves part (1). To prove part (2), observe that Lemma 3.17 implies that we can write v and w as v = ^¿Si YiUi and w = ^¿£w) ¿¿zj for some («4,..., wc(v)) e U and (zi,..., zc(w)) e W, and for some fixed Fqo-bases {7i, ..., 7C(v)} and {¿1,..., ¿c(w)} of Dv and Dw, respectively. Since Dw < Dv then c(w) < c(v), and we can extend {¿1,..., ¿c(w)} to an Fqo -basis {¿1,..., ¿c(v)} of Dv, and (z1,..., zc(w)) to (z1,..., zc(v)) e U. Set x := ¿¿zj and y := ^¿Si ¿¿yj, where yj := zj+i - zj if 1 < i < c(w) - 1, yc(w) := zi, and yj := -zj if c(w) + 1 < i < c(v). Then (yi,... ,yc(v)) e U and Dx = Dy = Dv, so by Lemma 3.17 we have x, y e vGL(n'qo) C vGo. Therefore x + y e vGo + vGo. Now Dw = Dx+y, so applying Lemma 3.17 again we get w e (x + y)GL(n,qo) C vGo + vGo. Thus (2) holds. □ Proof of Proposition 3.16. Suppose that r-1 < c(v) < r. Observe that n(r -1) = n(r) = 1, so for either value of c(v) we have vL = {u e V | c(u) = c(v)}, which in turn implies that vGo = vL. If c(v) = r then Dv = Fq, and clearly Dw < Dv for any w e V# \ vGo. So w e vGo + vGo by part (2) of Lemma 3.20, and thus V# \ vGo C vGo + vGo. Therefore diam(r) = 2. Now suppose that c(v) = r - 1, and let w e V # \ vGo. If c(w) < r - 1 then it follows from part (1) of Corollary 3.19 that Dw < ADv = for some A e F#. Thus w e (Av)Go + (Av)Go = vGo + vGo by Lemma 3.17. If c(w) = r let x := ^¿-i ajvj and y := /¿vi + Yvr, where {ai,..., ar-i} is an Fqo-basis of Dv, 7 e F# \ Dv, and = | aj+i - a if 1 < i < r - 3; 1 ai - ar-2 if i = r - 2. Then c(x) = c(y) = r-1 and c(x+y) = r, so x, y e vGo and w e (x+y)Go C vGo +vGo. Therefore V# \ vGo C vGo + vGo, and again we have diam(r) = 2. This completes the proof of part (1). If c(v) = 1 then we get the special case vL = vGo = (FqV0)#. Let distr(0V, w) denote the distance in r between the vertices 0V and w; we claim that distr(0V, w) = 158 Ars Math. Contemp. 13 (2017) 107-123 c(w) for any w G V. Let ¿(w) := distr(0V, w). Then w G Y by Proposition 3.18, where Y is as in (3.13), so w can be written as a sum of c(w) elements of (FqV0)# and thus ¿(w) < c(w). On the other hand w = ^¿=1 A^, where Aj G F# and uj G V0# for all i. Writing each uj as uj = J2"=i wj where G Fqo for all i, j, we get w = Ajwj where Aj = Aj^jj for each j. Hence Dw < (Ai,..., A£(w))f,0 , so that c(w) < ¿(w). Therefore ¿(w) = c(w), as claimed. It follows immediately that diam(r) = min {n, r}, and that diam(r) = 2 if and only if n = 2 or r = 2. This proves (2). Suppose that diam(r) = 2. Then c(w) < 2c(v) for any w G V# by part (1) of Lemma 3.20, and in particular 2c(v) > min {n, r} since there clearly exists u G V# with c(u) = min {n, r}. Hence c(v) < 1 min {n, r} implies that diam(r) > 2, and part (3) holds. Finally, let a := c(v), S := vGo, and n(a) as in (3.11). By Corollary 3.19 we have |S|< [ n 1 |GL (a,qo) ||F# : F# | s, L J q o where s is the largest divisor of d/n with s < n(a). Hence |S|2 + 1 1, where t = 17 if q0 = 2, and t = 0 if q0 > 3. Also, for q0 > 3, we have q0 - 1 > q^8, so that |F# : F# | < q0-5/8. With these bounds we obtain |S|2 + 1 < q0(an+r) + ki(qo) where k1(q0) is as defined in (4). It is easy to verify that if a < (r(n - 2) - k1(q0))/(2n) then 2(an + r) + k1(q0) < rn, so |S|2 + 1 < |V|, and thus diam(r) > 2 by Lemma 2.1. This proves part (4). □ Remark 3.21. Some small cases covered by Proposition 3.16 are summarised in Table 3.4.6. The cases left unresolved by Proposition 3.16 are the following: 1. 5 < r < n, r/2 < c(v) < r — 2; 2. 2 = n < r - 2, c(v) = 2; 3. 3 < n < r, maxn/2, (r(n - 2) - k1(q0))/(2n) < c(v) < r - 2. Let a := c(v) < r, S = vGo, and s as in Proposition 3.16 (4). Then s > 1, |F# : F#o|>qS-2 and n q o |GL (a,qc) | >q2a(n-1), so |2 |Go|2 + 1 > n ]?o |GL (a, qo) ||F# : F#01 ^ +1 > q2o(n-1) + 2(r-2) > qo . It is easy to show that if condition (1) or (2) holds then 2(a(n - 1) + r - 2) > rn, and thus |G012 + 1 > |V|. This, unfortunately, does not lead to any conclusion about diam(r). a C. Amarra et al.: Affine primitive symmetric graphs of diameter two 159 Table 3.4.6: r as in Proposition 3.16 for small values of r and n r n c(v) Conclusion about r = Cay(V, vGo ) 2 > 2 1 diam(r) = 2 by Proposition 3.16 (2) 2 diam(r) = 2 by Proposition 3.16 (1) 3 2 1 diam(r) = 2 by Proposition 3.16 (2) 2 diam(r) = 2 by Proposition 3.16 (1) 3 > 3 1 diam(r) = 3 by Proposition 3.16 (2) 2 diam(r) = 2 by Proposition 3.16 (1) 3 diam(r) = 2 by Proposition 3.16 (2) 5 2 1 diam(r) = 2 by Proposition 3.16 (2) 5 3 1 diam(r) = 3 by Proposition 3.16 (2) 5 4 1 diam(r) = 4 by Proposition 3.16 (2) 4 diam(r) = 2 by Proposition 3.16 (1) 5 > 5 1 diam(r) = 5 by Proposition 3.16 (2) 2 diam(r) > 2 by Proposition 3.16 (3) 4 diam(r) = 2 by Proposition 3.16 (1) 5 diam(r) = 2 by Proposition 3.16 (1) 3.4.2 Case H = rSp(n,q) By Theorem 2.5, Go = (GSp (n,qo) o Z,_i) x (r> and L = GSp (n, q0) o Zq-1. The main result in this section is parallel to part (4) of Proposition 3.16. Proposition 3.22. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with H = rSp(n, q) and i = 5. Then r is connected and G-symmetric if and only if r = Cay(V, vGo) for some v G V#. Moreover, if s := |t| = |G0 : L| and c(v) is as defined in (3.9), and if t :f 9/17 if qo = 2, :\l/2 if qo > 2 then the following hold: 1. If c(v) < 1 min {n, r} then diam(r) > 2. 2. If 3 < n < r, c(v) > n/2 and r > (n2 + n + 2st)/(n - 2), then diam(r) > 2. Proof. Assume that c(v) < 2min {n, r}. Let S = vGo, and let r' = Cay(V, vG°), such that G' satisfies Hypothesis 3.1 with H = rL(n, q) and i = 5. Then r is a subgraph of r', and hence diam(r) > diam(r'). If c(v) = 1 then diam(r') > min {n, r} > 2 by part (2) of Proposition 3.16, and if c(v) > 2 then diam(r') > 2 by part (3) of Proposition 3.16. In both cases diam(r) > 2. This proves statement (1). We now prove statement (2). Observe that for any A G F# and g G GSp (n, q0), we have Avg = vXg G vGSp(n,qo) if and only if A/„ G Zqo-1, the subgroup of scalar 160 Ars Math. Contemp. 13 (2017) 107-123 matrices in GL (n, q0). Hence vL = |JA£AvGSp(n'qo) can be written as a disjoint union vL = |JXeT AvGSp(n'qo), where T is a transversal of F#o in F#. Thus |vL| < |T||GSp (n,qo) | = (q0 - 1)|Sp (n, qo) | and |S| < s|vL|, where s = |G0 : L|. We have n/2 - „"2/4 IT 2i -i \ . Jn2+n)/2 |Sp (n, qo) | = q?^ (q2* - 1) < q0" Also, as in the proof of Proposition 3.16 (4), we have s < qg4 for any s, where t = 17 if q0 = 2, and t = 1 if q0 > 3. Hence |S|2 + 1 < s2(q° - 1)2q°2+° < q°2+°+2o+2si. If r > (n2 + n + 2st)/(n - 2) then rn > n2 + n + 2r + 2st, so |V| > |S|2 + 1 and diam(r) > 2 by Lemma 2.1. Therefore part (2) holds. □ 3.5 Class Ce In this case dim (V) = r4 where r is a prime different from p, q is the smallest power of p such that q = 1 (mod |Z(R)|) for some R in Table 2.3.5, and G0 = (Z,_i o R).T x (r>, with T as in Table 2.3.5. By Theorems 2.4 and 2.5, if H = rL(n, q) then R is of type 1 or 2, and if H = TSp(n, q) with q odd then R is of type 4. Proposition 3.23 is an extension of [2, Proposition 3.6], and is proved somewhat similarly. Proposition 3.23. Let V and G0 be as above, and let r := Cay(V, S) for some G0-orbit s c v #. 1. Suppose that r is odd, q = 1 (mod r), and R is Type 1. If diam(r) = 2 then 1 < t < 3, r < r0(t), and q < q0(r, t), where r0(t) and q0(r, t) are given in Table 3.5.7. 2. Suppose that r = 2, t > 2, q = 1 (mod 4), and R is Type 2. If diam(r) = 2 then 2 < t < 6 and q < q0(t), where q0(t) is given in Table 3.5.8. 3. Suppose that r = 2, t > 2, q is odd, and R is 7ype 4. If diam(r) = 2 then 2 < t < 7 and q < q0(t), where q0(t) is given in Table 3.5.9. 4. Suppose that r = 2, t = 1, q is odd, and R is 7ype 2 or 4. Then diam(r) = 2 for any S. Proof. If q = p£ and R is Type 1 or 2, then |G0| = ¿(q - 1)r2t|Sp (2t, r) | < ¿(q - 1)r2t2+3t. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 161 Table 3.5.7: Bounds for r and q when R is Type 1 t 1 2 3 r0(t) 11 3 3 q0(3,t) 186619 73 11 q0(5,t) 521 - - q0(7,t) 71 - - q0(11,t) 23 - - Table 3.5.8: Bounds for q when R is Type 2 and t > 2 t 2 3 4 56 q0(t) 23029 569 73 17 5 Table 3.5.9: Bounds for q when R is Type 4 and t > 2 t 2 3 4 567 q0(t) 1913 149 37 11 5 3 Suppose first that R is Type 1. In this case r is odd and q = pe = 1 (mod r), so I < r - 1, q > r, and |Go|2 + 1 < ((q - 1)r2i2+3i+!)2 + 1 < q4t2+6i+4. It can be shown that 4t2 + 6t + 4 < r* for the following cases: t > 5 and r > 3, t = 1 and r > 17, t = 2 and r > 7, and t G {3,4} and r > 5. Thus for all these cases |G0|2 + 1 < |V |. For all remaining pairs (r, t) define n(q,r,t) := ((r - 1)(q - 1)r2t|Sp (2t, r) |)2 + 1 - qr'. Then |Go|2 + 1-|V| < n(q,r,t) andn(q,r,t) < 0if q > ((r - 1)r2t|Sp (2t,r) |)2/(r'-2). Getting the largest prime power q = pe = 1 (mod r) less than or equal to this bound, with I < r - 1 and n(q, r, t) > 0, gives the values q0(r, t) in Table 3.5.7, and for each t we take r0(t) to be the largest value of r for which there exist such q. In particular, n(q, r, t) < 0 for the following cases: (r, t) = (13,1) and q > 13, (r, t) = (5, 2) and q > 7, (r, t) = (3,4) and q > 3; for these cases there is no value of q less than or equal to the given bound that satisfies all the required conditions. This proves part (1). Now suppose that R is Type 2 with t > 2. Then r = 2 and q = p/ = 1 (mod 4), so I < 2, q > 4, and |Go|2 + 1 < ((q - 1)22t2+3t+1 )2 + 1 < q2t2+3t+3. We have 2t2 + 3t + 3 < 2* whenever t > 7, hence |G0|2 + 1 < |V| for all such t. For t g {1,..., 6} define n(q,t) := (2(q - 1)22t|Sp (2t, 2) |)2 + 1 - q2', 162 Ars Math. Contemp. 13 (2017) 107-123 and observe that |Go|2 + 1 — |V| < n(q,t) < 0 for all q > (22i+1|Sp (2t, 2) |)1/(2' 1 1). The values of q0(t) in Table 3.5.8 are the largest prime powers q = pe = 1 (mod 4) less than or equal to these bounds, with I < 2 and satisfying n(q, t) > 0. This proves (2). For (3), suppose that R is Type 4 with t > 2. Then r = 2 and |Z(R)| = 2, so I = 1 and q = p. Also q > 3, so q3/2 > 4. We have |Go| = (q — 1)22t |O- (2t, 2) I < (q — 1)22i2+i+2 so |Go|2 + 1 < ((q — 1)22i2+i+2)2 + 1 < q242t2+4+2 < q3t2 +3 We have 3t2 + §1 + 5 < 24 (and hence |G012 + 1 < | V|) for all t > 8. For t e {2,..., 7} define ( | |) n(q,t) := ((q — 1)22t |O- (2t, 2)|)2 + 1 — /. Then |Go|2 + 1 — |V| < n(q,t) < 0 for all q > (22t |O- (2t, 2)|)1/(2'-1-1). As in the previous cases we take q0(t), 2 < t < 7, to be the largest prime q less than or equal to these bounds such that n(q, t) > 0. This yields Table 3.5.9 and proves (3). Statement 4 for the case where R is type 2 is precisely [2, Proposition 3.6 (2)]. For the case where R is type 4 define the matrices a, c e GL (V) by —01 0) and c := (7 —7 where 7,7 e Fq such that 72 + 72 = —1. Then (a, c) is a representation of R in GL (2, q) (see [6, pp. 153-154]). Since R is irreducible on V, any R-orbit vR in V# contains a basis {v1, v2} of V, and vG° contains (v1}# U (v2}#. Clearly V# C (v1}# + (v2}#. Therefore V C vG° + vG°, and thus diam(r) = 2. This proves (4), and completes the proof of the proposition. □ 3.6 Class C7 In this case V = ®f=1Uj with Uj = F^ for all i, m > 2, t > 2, and d = m4. Assume that B is a tensor product basis of V, with B := {®t=1uj,j 11 < j < m} . the C4 case, it is not d have As in the C4 case, it is not difficult to show that for any v = J21=1 (®j=1 vj,j) e V# we i=i where t acts on each Uj with respect to the basis {«¿j | 1 < j < m}. 3.6.1 Case H = TL(n, q) By Theorem 2.4 Go = (GL (m,q) Sym (t)) x (t>. (3.15) If t = 2 then we obtain the examples in Proposition 3.13 with k = m. We state this in the next corollary, which is analogous to [2, Corollary 3.7]. C. Amarra et al.: Affine primitive symmetric graphs of diameter two 163 Corollary 3.24. Let V = g^F^ and let G0 be as in (3.15) with m > 2 and t = 2. Then the G0-orbits in V # are the sets Ys for each s € {1,..., m}, where Ys is as defined in (3.7). Moreover, for any G0 -orbit S C V #, the graph Cay(V, S) has diameter 2 if and only if S = Ys for some s > m/2. Proof. This follows immediately from Lemma 3.12 and Proposition 3.13. □ Using Lemma 2.1, we get the following bounds which significantly reduce the cases that remain to be considered. It turns out that these are exactly the same as those in [2, Proposition 3.8]; we prove them here for subgroups of rL(n, q). Proposition 3.25. Let r be a graph and let G < Aut (r), such that G satisfies Hypothesis 3.1 with G0 as in (3.15), m > 2 and t > 3. Then r is connected and G-symmetric if and only if r = Cay(V, vGo) for some v € V#. Moreover, if diam(r) = 2 then either: 1. m = 2 and t € {3, 4, 5}; or 2. t = 3 and m € {3, 4, 5}. Proof. Recall that (agi) g g2 g • • • g gt = gi g • • • g (ag) g • • • g gt for all gi,..., gt € GL (m, q), so that |Go| < |GL (m, q) |t t! i(q - 1)-(t-i). Now |GL (m, q) | < qm(m—i)qm—i(q - 1) = qm2-i(q - 1), s < qs-i for all s > 2 and q > 2, and i < p = q for all i > 1 andp > 2, so that |Go|2 + 1 < (q(m2-i)t(q - 1)t)2 (q2t(t-i))2 q2(q - 1)-2(t-i) < qt2 + (2m2-3)t+4. It can be shown that t2 + (2m2 - 3)t + 4 < mt whenever t > 7 and m > 2, and whenever t € {3,4,5, 6} and m > m0(t), where m0(t) is as given in Table 3.6.10. Hence |G0|2 + 1 < |V| for all such pairs (m,t). Of the remaining pairs we can eliminate (2, 6) and (6,3) by considering n(q, m, t) := (t!)2q2t(m -i)+4 - qmt; it can be shown that n(q, 2, 6) < 0 for all q > 2 and n(q, 6, 3) < 0 for all q > 7. For q € {2, 3,4,5} it can be checked that 36i2 |GL (6, q) |6(q - 1)-4 + 1 < q2i6. Therefore |G0|2 + 1 < |V| if (m, t) € {(2, 6), (6, 3)}, which completes the proof. □ Table 3.6.10: Values for m0(t) t 3 4 5 6" m0(t) 6 2 2 2 3.6.2 Case H = rSp(n, q) By Theorem 2.5, both q and t are odd and G0 = (GSp (m, q) fa Sym (t)) x (r>. (3.16) Hence q, t > 3. 164 Ars Math. Contemp. 13 (2017) 107-123 Proposition 3.26. Let r be a graph and G < Aut (r) such that G satisfies Hypothesis 3.1 with G0 as in (3.16), m > 2 and t > 3. Then r is connected and G-symmetric if and only if r = Cay(V, vGo ) for some v e V#. Moreover, if diam(r) = 2 then either: 1. m = 2 and t e {3, 5}; or 2. t = 3, m = 4, and q = 9. Proof. In this case |G0| < |GSp (m, q) 1! £(q - 1)-(t-1), where |GSp (m, q) | = (q - 1)Sp (m, q) < (q - 1)q1 (m2+m). Also s < ks/2 for all k > 3 and s > 2, so that £ < q, t! < q4(t-1)(t+2), and |G0|2 + 1 < (q - 1)2iqi(m2+m)+1 (t-1)(t+2) + 1(q - 1)-2(t-1) < q 2 i2 + (m2+m+ 2 )i+2. It can be shown that 112 + (m2 + m + 1) t + 2 6 and m > 2, t = 3 and m > 5, and t = 5 and m > 3. So |G012 + 1 < |V| for all such pairs (m, t). Let n(q, m, t) := (t!)2qt(m2+m)+3 - qm. If (m, t) = (4,3) then for all q > 37 we get |G012 + 1 -|V| 2. Proof. In the following we will make repeated use of the fact that for u,v,w,x,y, z G Z|, if u + w = y, v + w = z and v + x = y, then u + x = z. Let k > 2 and suppose that (W,B) is a spherical latin bitrade whose canonical group is isomorphic to Z|. So, by Theorem 1.3, both W and B embed in Z|. Recall that we may assume that the row, column and symbol sets of W (and of B) are pairwise disjoint; denote them, respectively, by R = {r1,r2,..., re}, C = {c1, c2,..., cm} and S = {si, s2,..., sn}. Let GW,B be the related triangulation and G be the underlying graph of this triangulation. As GW,B has a proper face 2-colouring, G is Eulerian, and, as (W, B) is a latin bitrade, the minimum degree of G is at least four. Moreover, GW,B is a triangulation of the sphere, so, by Euler's formula, G contains at least six vertices of degree four. As spherical latin bitrades in the same main class all have isomorphic canonical groups, without loss of generality, we may assume that the degree of r1 is four, and (r1,c1,s1), (r1, c2, s2) G B and (r1, c1, s2), (r1, c2, s^ G W where c1 = c2 and s1 = s2. Hence, as (W, B) is a latin bitrade, there exist x1, x2,x3, x4 G R \ {r^ such that (x1,c2, s1), (x3,c1, s2) G B and (x2,c1 ,s1), (x4,c2, s2) G W (see Figure 2 for an illustration of the corresponding faces). As W embeds in Z§, x2 = x4 and, as B embeds in Z§, x1 = x3. Suppose that x1 = x2. 172 Ars Math. Contemp. 13 (2017) 107-123 Let W' = {(ri, ci, S2), (ri, C2, si), (xi, ci, si), (xi, C2, S2)} and B' = {(ri, ci, si), (ri,C2, S2), (xi, ci, S2), (xi,C2, si)}. Then (W', B') is a spherical latin bitrade such that W' C W and B' C B. As (W, B) is connected, it must be the case that W' = W and B' = B. However, the canonical group of (W', B') is Z2, a contradiction. So xi = x2; in which case G contains a subgraph H = (V, E) where V = {ri, xi, x2, ci, si, s2} and E = {rici, risi, ris2, xici, xisi, xis2, x2ci,x2si,x2s2}. However, H is isomorphic to K3 3; which contradicts GW,B being a spherical embedding. □ 2 Existence results 2.1 Directed Eulerian spherical embeddings Let D be a, not necessarily simple, digraph of order n with vertex set V (D) = {vi, v2,..., vn}. The adjacency matrix A = [aj ] of D is the n x n matrix where entry ajj is the number of arcs from vertex vj to vertex Vj. The asymmetric Laplacian of D is the n x n matrix L(D) = B - A where B is the diagonal matrix in which entry bn is the out-degree of vertex Vj. The digraph D is said to be Eulerian if, for each v G V(D), the out-degree at v equals the in-degree at V. Hence, in an Eulerian digraph we will simply refer to the degree of a vertex v, i.e. deg v. Let D be a connected Eulerian digraph of order n with vertex set V(D) = {vi, v2,..., vn}. Fix an i, where 1 < i < n and define L'(D,i) to be the matrix obtained by removing row and column i from L(D). As D is connected and Eulerian, the group Zn-i/L'(D, i)Zn-i is invariant of the choice of i, see [17, Lemma 4.12]. Hence, the abelian sandpile group of the connected Eulerian digraph D can be defined to be the group S(D) = Zn-i/Zn-iL'(D,n); moreover S(D) = Zn-i/Zn-iL'(D, i), for any 1 < i < n. Consider an embedding D of a connected Eulerian digraph D in an orientable surface S. If each face of the embedding corresponds to a directed cycle in D, equivalently the rotation at each vertex alternates between incoming and outgoing arcs, then the embedding is said to be a directed Eulerian embedding, see [2, 3]. If the embedding is in the sphere we call it a directed Eulerian spherical embedding. Suppose that G is a face two-coloured triangulation of the sphere. By [16], the underlying digraph of G has a vertex three-colouring with colour classes R, C and S. Tutte [21] described a construction, from G, of directed Eulerian spherical embeddings D/(G) = D/ with vertex set I, where I G {R, C, S}. We give a description of the construction from [19]. Let {I, Ii,/2} = {R, C, S}. Consider a vertex vj g I. Then vj has even degree, say d, the rotation at i is (ui, vi, u2, v2,..., ud/2, vd/2), where, without loss of generality, Uj G Ii and vj G 12 for all 1 < j < d/2 and the edge ej between Uj and vj in the rotation is contained in a black face. Then in D/ there are d/2 outgoing arcs from vertex vj, say aj, 1 < j < d/2, one for each black face, and the terminal vertex for arc aj is the vertex in I contained in the white face containing edge ej. Clearly, D/ inherits a spherical embedding from G in which the arc rotation at each vertex alternates between incoming and outgoing arcs, so D/ has a directed Eulerian spherical embedding. As the sphere is connected the K. Bonetta-Martin and T. A. McCourt: On which groups can arise as the canonical group 173 graph underlying DI is connected. Note that given any of DR, DC or DS the original face two-coloured triangulation can be obtained by reversing the above construction: Lemma 2.1 (Tutte, [21]). Given a directed Eulerian spherical embedding D, there exists a face 2-coloured spherical triangulation G with a vertex 3-colouring given by the vertex sets R, C and S, such that for some I G {R,C, S}, DI (G) = D. Tutte's Trinity Theorem [21] states that |S(Dr)| = |S(Dc)| = |S(Ds)|. For a spherical latin bitrade (W, B) with corresponding face two-coloured triangulation G, this result was strengthened implicitly in [1] and explicitly in [19] to S(DR) = S(DC) = S(DS) = Aw = Ab . Given an arbitrary directed Eulerian spherical embedding applying the above construction in reverse yields a face two-coloured triangulation. However, the underlying graph is not necessarily simple. In order to make use of the above equivalences (between sandpile groups and canonical groups of spherical latin squares) we make use of the following result. Proposition 2.2 (McCourt, [19]). Suppose that D is a directed Eulerian spherical embedding with underlying digraph D. Further suppose that D is connected, has no loops, no cut vertices and its underling graph has no 2-edge-cuts. Then there exists a spherical latin bitrade whose canonical group is isomorphic to S(D). Hence, in order to construct a spherical latin bitrade with canonical group r it suffices to find a directed Eulerian spherical embedding satisfying the connectivity conditions of Proposition 2.2 whose abelian sandpile group is isomorphic to r. 2.2 Arbitrary rank In this section we will construct families of canonical groups that have arbitrary rank. We will make repeated use of the following, elementary lemma. Lemma 2.3. Let 2 < p,a and 0 < x,y, £. Further let r = p(x + 1) + a — x — 1, s = p(y + 1) + a — y — 1 and ti,j G Z, for 1 < i < m and 1 < j < £. Then the matrix p —p + 1 0 ••• 0 0 ••• 0 —1 0 • 0 —p r —p ••• —p 0 ••• 0 x + 1 — a 0 • 0 0 —p + 1 0 ••• 0 — 1 0 • 0 pIx 0 —p + 1 0 ••• 0 — 1 0 • 0 0 —1 0 ••• 0 —p +1 0 • 0 ply 0 —1 0 ••• 0 —p +1 0 • 0 0 y + 1 — a 0 ••• 0 —p ... —p s t1,1 • • h,e 0 —1 0 ••• 0 0 ... 0 —p +1 *2,1 • • t2,e 0 0 0 ••• 0 0 ••• 0 0 t3,1 • • h,e 0 0 0 ••• 0 0 ••• 0 0 ¿m, 1 • • tm,t L 174 Ars Math. Contemp. 13 (2017) 107-123 reduces (under row and column operations invertible over Z) to 1 0 0 ••• 0 0 0 • • 0 0 ap 0 ••• 0 0 0 • • 0 0 0 0 0 • •0 pIx+y 0 0 0 0 • •0 0 0 0 ••• 0 p t1,1 • • t1,£ 0 0 0 ••• 0 -p t2,1 • • t2,£ 0 0 0 ••• 0 0 ts,1 • • t3,£ 0 0 0 ••• 0 0 tm,1 • • tm,£ Proof. For 1 < i < x and 1 < j < y add Row 2 + i and Row 2 + x + j to Row x + y + 3 of L. Subsequently, for 1 < i < x, add Column 2 + i to Column 2 and, for 1 < j < y, Column 2 + x + j to Column 3 + x + y. Next add Column 2 to Column 1. Now add Column 1 to Column 3 + x + y and p - 1 copies of Column 1 to Column 2. Row 1 can now be used to clear all non-zeros from Column 1. Once this is completed it is easy to see that the remaining non-zeros in Column 2 can also be cleared. □ The proof of Lemma 2.4 is essentially a special case of the proof of Theorem 2.6, however, to aid the reader, we detail this simpler case before proving the general result. Lemma 2.4. Let 1 < k and let 2 < m, a1, a2,..., ak. Then there exists a spherical latin bitrade whose canonical group is isomorphic to 0k=1 Zma.. Proof. We begin by defining a digraph Dm;ai,a2,...,ak with vertex set {a0, a1,a2,..., ak, 71,72,..., Yk } and • for each 1 < i < k: o m - 1 arcs from a4 to Yi and m - 1 arcs from Yi to a^ o aj - 1 arcs from ai-1 to Yi and a4 - 1 arcs from Yi to ai-1; o an arc from aj to ai-1; • for each 1 < i < k - 1: an arc from y% to Ym; and • an additional arc from a0 to y1 and an additional arc from Yk to ak. The digraph Dm;ai,a2,...,ak has a directed Eulerian spherical embedding and satisfies the connectivity conditions of Proposition 2.2, as can be seen from Figure 3 (in this figure t arcs from u to v alternating with t arcs from v to u are represented by a bidirectional edge labelled t). Hence, there exists a spherical latin bitrade whose canonical group is isomorphic to S {Dm;ai,a2,...,ak ). Suppose that we order the vertices of Dm;ai,a2,...,ak by ak,Yk, ak-1, Yk-1,..., a2, Y2, a1, y1 , a0, and construct the associated asymmetric Laplacian. Then, removing the row and column corresponding to a0 yields the reduced asymmetric Laplacian L'(Dm;ai,a2,...,ak). m —m + 1 —m m + ai — 1 Let k > 1 and m, a1, a2,..., ak+1 > 2. Note that £'(Dm;ai) = reduces to 1 0 0 mai ; so S(Dm;ai ) = Zm Assume that S(Dm;ai,a2,...,ak) is isomorphic to 0, 1 < i < k, the reduced asymmetric Laplacian L'k = L'(D Z 1 ^ ma . Setting ai - 1 = aj for m;ai ,a2,...,ak ) is shown below. K. Bonetta-Martin and T. A. McCourt: On which groups can arise as the canonical group 175 ak-2 ak-1 Figure 3: A directed Eulerian spherical embedding of D m;ai,a2,...,ak" Lk = m —m 0 0 0 0 0 0 —m + 1 m + a'k —a'k —1 0 0 0 0 —1 0 —ak 0 m + ak —m + 1 —m + 1 m + a;k-1 0 —ak-i 01 —m + 1 0 0 0 0 —m + 1 m + ak-2 0 0 m + a2 —m + 1 —m +1 m + a1 Now, consider the digraph Dm;ai,a2,...,afc+1. Applying Lemma 2.3, with p = m and x = y = 0, to rows ak+1,7k+1, ak,7k we have that Ck+1 = L'(Dm;a1,a2,...,ak+1) reduces to 1 0 0 ••• 0 0 mak+1 0 ••• 0 0 0 0 0 c Lk It follows that S(D 1) is isomorphic to 0 k=i11 Z. m;ai ,a2 ,...,ak+ □ 1 k1 k1 It is now easy to establish the existence of spherical latin bitrades whose canonical groups can be expressed as the direct sum of components of composite order. Theorem 2.5. Suppose that r is a group isomorphic to a direct sum of cyclic groups of composite order; i.e. r is isomorphic to ©k=1Z„., where each n is composite. Then there exists a spherical latin bitrade whose canonical group is isomorphic to r. Proof. Let n1, n2,..., nk be composite integers and consider r = ©k=1Zni. Recall that if gcd(n„, nv) = 1, u = v, then ©k=1Z„i ^ Zrn © • • • © Z„u_i © Z„u+i © • • • © Zn„_i © Z„v+1 © • • • © Z„fc © Z„u„v. Thus we may assume that gcd{n1, n2,..., nk} = 1. Hence there exists a prime, p say, such that p divides gcd{n1, n2,..., nk}. Note that, as n is composite for all 1 < i < k, p = n4. By setting m = p and applying Lemma 2.4 the result follows. □ The next result addresses the existence of spherical latin bitrades for which the Smith Normal Form of their canonical groups contains components of prime order. 176 Ars Math. Contemp. 13 (2017) 107-123 Theorem 2.6. Let p be a prime and let 2 < aL,a2,..., ak. Further let n < 1 + 2E j = i(ai - 1). Then there exists a spherical latin bitrade w1iose canomcal gmup is isomorphic to Znp e ® Z Vi=1 pai Proof. If n = 0, then this is Lemma 2.4. So for the remainder of the proof assume that n > 1. As n < 1+2Ek=1(ai-1) thereexistsak', 0 < k' < k,andt, 0 < t < 2(ak'+1-1) such that k' n =1 + 2 ^(ai - 1)+ t. First construct the graph Dp;ai,a2, following vertices, (from the proof of Lemma 2.4). Next, add the • for each 1 < i < k': add vertices 5ij and for all 1 < j < ai - 1; • for each 1 < j < [t/2]: add vertices 5k'+1jj; • for each 1 < j < |i/2j: add vertices ek'+1j-; and • the vertex e10. Now add arcs to and replace arcs from D p;ai,a2, ,Ofc as illustrated in Figures 4 and 5. Note that D has a directed spherical embedding, and that it satisfies the connectivity conditions of Proposition 2.2. Therefore, there exists a spherical latin bitrade whose canonical group is isomorphic to S (Dp; ). For ease of notation, let a — 1 for 1 < i < k' d = ^ [t/2] for i = k' + 1 0 otherwise and a — 1 for 1 < i < k' |_t/2j for i = k' + 1 0 otherwise Suppose that we order the vertices of Dp by (ak, Yk , 5k,dfc , . . . , 5k,1, £k,ek , . . . , . . . , (a2 , Y2 , 52,d2 , . . . , 52,b £2,e2 , . . . , ^2,1^ ^^ 7l, 52,di , . . . , £1,ei, . . . , £1,1, ^1,0^ a0 and construct the associated asymmetric Laplacian. Then, removing the row and column corresponding to a0 yields the reduced asymmetric Laplacian L'(D n p;a1,»2, k+1. Let k > 1 and p, a1, a2,..., aj+1 > 2 and let 1 < n < 1 + 2J2¿=1. (ai — 1). Then, letting x = dL, y = eL and r = p(x + 1) + aL — x — 1, L ^Dmin{n,1+2(ai-1)}j p —p +1 0 ••• 0 0 ... 0 0 —p r —p ... —p 0 ... 0 0 0 —p + 1 oId1 0 ••• 0 0 0 —p + 1 0 ••• 0 0 0 —1 0 ••• 0 0 plei 0 0 —1 0 ... 0 0 —1 0 ... 0 0 ••• 0 p e K. Bonetta-Martin and T. A. McCourt: On which groups can arise as the canonical group 177 Let a = ai — 1. For k' > 0: For k' = 0, if t = 21. For k' = 0, if t = 2^ +1: Figure 4: Constructing Dm;a1,a2 ak for arcs incident with a0. p -1 p -1 p -1 p -1 p -1 p -1 178 Ars Math. Contemp. 13 (2017) 107-123 For 1 < i < k': Yi-i ai-i Let a = ak'+i — 1, then, if t = 2i: Yk'+i Yk'+i ak'+i a^'+i Again let a = ak'+i — 1, then, if t = 2i + 1: Yk'+i Yk'+i ak'+i ak'+i Figure 5: Constructing Dm;a a2,...,ak for arcs not incident with a0. Y Y p -1 p -1 p -1 p -1 a p -1 p -1 p -1 p -1 K. Bonetta-Martin and T. A. McCourt: On which groups can arise as the canonical group 179 Which reduces, under a similar argument to that used to prove Lemma 2.3, to 1 0 0 ••• 0 0 pa1 0 ••• 0 0 0 p!d1+e1 + 1 0 0 Hence, S(DP^T^2^1-1^) = Zmin{n'1+2(ai-1)} ,min{n,1+2 Ej=1(ai-1)}^ ^ ^ Assume that S [Dp-!a1{a2,...,ak © Zpa1. in{n,1+2£ J=1(ai-1)} © © k = 1 Zpa, Denote L (dP^^F^^ 1)}) by Ck = [¿j]. Let x = 4+1, y = efc+1, p(x + 1) + ak+1 = dk+1 - 1 and s = p(y + 1) + ak+1 - y - 1. Then the asymmetric Laplacian is CJ Dn L 1 Dp;a 1,a2 ,afc+1 r p -p +1 0 ... 0 0 • •• 0 -1 0 • • 0 ] -p r -p • • • -p 0 • •• 0 x +1 - ak+1 0 • • 0 0 -p + 1 pIx 0 • •• 0 -1 0 • • 0 0 -p + 1 0 • •• 0 -1 0 • • 0 0 -1 0 • •• 0 ply -p + 1 0 • • 0 0 -1 0 • •• 0 -p + 1 0 • • 0 0 0 y +1 - ak+1 -1 0 0 • •• 0 • •• 0 -p 0 ... -p ... 0 s -p + 1 ¿1,2 • ¿2,2 • • ¿1,2k+n • ¿2,2k+n 0 0 0 • •• 0 0 • •• 0 0 ¿3,2 • • ¿3,2k+n 0 0 0 • •• 0 0 • •• 0 0 ¿2k+n,2 • • ¿2k+n,2k+n . Applying Lemma 2.3 to rows ak+1,7k+1, ¿k+1,; Yk of Lk+1(Dn;a1,a2,...,afc+1 ) reduces it to > ^k+1,1, £k+1,y, . . . , ek+1,1,ak, 1 0 0 ••• 0 0 ••• 0 0 pak+1 0 ••• 0 0 ••• 0 0 0 0 ••• 0 pIx+y 0 0 0 ••• 0 0 0 0 ••• 0 c 0 0 0 ••• 0 Therefore S(Dn;a1,a2,...,afc+1) = ^n © Z 2.3 Canonical groups of rank two □ In this section we will restrict our attention to canonical groups of rank two. We show that, with two exceptions and a further two possible exceptions, any finite abelian group of rank two is isomorphic to the canonical group of some spherical latin bitrade. 180 Ars Math. Contemp. 13 (2017) 107-123 We will make use of the following elementary lemma. Lemma 2.7. Let 2 < d, 1 < x, 2 < y and U,j G Z for 1 < i < x and 1 < j < y. Further let M = [mj] be the d — 1 by d matrix where ( 2 if i = j mj = < —1 if j = i +1 or j = i — 1 . [ 0 otherwise Then the d + x — 2 by d + y — 2 matrix 0 • 0 M 0 • 0 0 • 0 0 • 0 t1,1 tl,2 tl,3 • • ti,v 0 • 0 tx,1 tx,2 tx,3 • • tx,y reduces (under operations invertible over Z) to Id-2 0 0 0 • 0 0 0 0 • 0 0 0 d 1 — d 0 • 0 0 0 t1,1 t1,2 t1,3 • t1,y 0 0 tx,1 tx,2 tx,3 tx,y Proof. When d =2, the result is trivial. Assume that the statement holds for d = k, and consider L(k + 1). Then L(k +1) reduces to 0 0 0 0 0 Ik-2 0 0 0 0 0 0 0 k 1 — k 0 0 0 0 0 —1 2 —1 0 0 0 0 0 t1,1 t1,2 t1,3 t1,y 0 0 0 tx,1 tx,2 tx,3 tx,y Adding k — 1 copies of Row k to Row k — 1 followed by adding one copy of the updated Row k — 1 to Row k yields a 1 in entry (k — 1, k — 1) and this is now the only non-zero in Column k — 1. The result follows. □ Lemma 2.8. Suppose that 1 < a,b, c. Then there exists a spherical latin bitrade whose canonical group is isomorphic to Zab+bc+ac+i © Za b+bc+ac+1. K. Bonetta-Martin and T. A. McCourt: On which groups can arise as the canonical group 181 a a Y c Figure 6: A directed Eulerian spherical embedding of Da,b,c Proof. Without loss of generality we may assume that 1 < a < b < c. Define Da,b,c to be the digraph of order a + b + c +1 with vertex set {a1, a2,..., aa, P1, P2,..., Pb, Yi, Y2, ...,Yc,S} and • for 1 < i < a — 1 an arc from a.i to ai+1 and an arc from ai+1 to a4; • for 1 < i < b — 1 an arc from pi to Pi+1 and an arc from pi+1 to Pi; • for 1 < i < c — 1 an arc from Yi to Yi+1 and an arc from Yi+1 to Yi; • for each i g {a, P, y} an arc from S to i1 and from i1 to S; and • a arcs from pb to Yc and from yc to Pb; b arcs from aa to Yc and from yc to aa; and c arcs from aa to Pb and from pb to aa. Note that Da b c has a directed Eulerian spherical embedding, see Figure 6, and that Da b c satisfies the connectivity conditions of Proposition 2.2. Hence, there exists a spherical latin bitrade whose canonical group is isomorphic to S(Da,b,c). Suppose that we order the vertices of Dajbjc by Y1,Y2,...,Yc-2,Yc-1,Yc, P1, P2,.. ., Pb-2 ,Pb-1,Pb,a1,a2, .. .,aa-2,aa-1,aa,S. Let L'(Da,b,c) be the reduced asymmetric Laplacian for Da,b,c obtained by removing the row and column corresponding to S. When a = b = c =1, £'(D1iM) = 3 -1 -1 -1 3 -1 -1 -1 3 , which reduces to 1 0 0 0 4 0 0 0 4 Suppose that 2 < a,b, c. Consider L'(Dabc), via three applications of Lemma 2.8 and setting a + b + c +1= t, this reduces to 182 Ars Math. Contemp. 13 (2017) 107-123 0 0 0 ••• 0 0 0 0 • • • 0 0 0 Ic-2 0 0 0 ••• 0 0 0 0 • • • 0 0 0 0 • • • 0 c 1-c 0 ••• 0 0 0 0 • • • 0 0 0 0 • • • 0 -1 t - c 0 ••• 0 0 —a 0 • • • 0 0 —b 0 • • • 0 0 0 0 0 0 • • • 0 0 0 Ib-2 0 • • • 0 0 0 0 0 0 • • • 0 0 0 0 • • • 0 0 0 0 ••• 0 b 1—b 0 • • • 0 0 0 0 • • • 0 0 —a 0 ••• 0 —1 t—b 0 • • • 0 0 —c 0 • • • 0 0 0 0 ••• 0 0 0 0 0 Ia-2 0 • • • 0 0 0 0 ••• 0 0 0 0 0 0 • • • 0 0 0 0 ••• 0 0 0 0 • • • 0 a 1—a 0 ••• 0 0 —b 0 ••• 0 0 —c 0 • • • 0 —1 t — a Computing the Smith Normal form of c 1 —c 0 0 0 0 —1 t —c 0 —a 0 —b 0 0 b 1 —b 0 0 0 —a —1 t —b 0 —c 0 0 0 0 a 1 —a 0 -b 0 c 1 t a we have that S(D0,6,c) = Zab+bc+ac+i © Zab+bc+ac+i. The cases where 1 = a 2, with two exceptions and a further two possible exceptions, there exists a spherical latin bitrade whose canonical group is isomorphic to Zn © Zm. The exceptions are as follows. There does not exist a spherical latin bitrade with canonical group isomorphic to Z2 © Z2 or Z3 © Z3. There may or may not exist a spherical latin bitrade with canonical group isomorphic to Z5 © Z5 or Zr © Zr for some r greater than 1011. Finally, if we assume the Generalised Riemann Hypothesis, then there exists a spherical latin bitrade with canonical group isomorphic to Zr © Zr. Proof. If n and m are coprime, then Zn © Zm = Znm and the result follows from [9] (it also follows from Lemma 2.4 with k = 1). So assume that n and m are not coprime; that is, we are in the rank 2 case. Suppose that n = m. If n and m are both composite, then the result follows from Theorem 2.5. So suppose that n is prime and m is composite. Then as n and m are not coprime m = kn for some k > 1 and the result follows from Theorem 2.6. So, suppose that n = m. If there exist a, b, c > 1 such that ab + ac + bc +1 = n, then by Lemma 2.8 there exits a spherical latin bitrade whose canonical group is isomorphic to Zn © Zn. In [4] Borwein and Choi proved that there are at most nineteen integers that are not of the form ab + ac + bc +1 where a,b,c > 1. The first eighteen are: 2, 3, 5, 7, 11, 19, 23, 31, 43, 59, 71, 79, 103, 131, 191, 211, 331 and 463. The nineteenth is greater than K. Bonetta-Martin and T. A. McCourt: On which groups can arise as the canonical group 183 ai am-i —• <> a m m Figure 7: Directed Eulerian spherical embedding of a digraph with abelian sandpile group isomorphic to Z6m+5 © Z6m+5, when m € {1,3,9,11, 21, 31}. Figure 8: Directed Eulerian spherical embedding of a digraph with abelian sandpile group isomorphic to Z3m+1 © Z3m+i, when m € {2,6,10,14,26,34,70,110,154}. 1011 and is not an exception if the Generalised Riemann Hypothesis is assumed. For n G {7,11,19, 23, 31,43, 59, 71, 79,103,131,191, 211,331,463} directed Eulerian spherical embeddings whose underlying digraphs satisfy the connectivity conditions of Proposition 2.2 and with abelian sandpile groups isomorphic to Zn © Zn are given in Figures 7 and 8.2 Cavenagh and Wanless noted in [9] that there does not exist a spherical latin bitrade with canonical group isomorphic to Z2 © Z2 (also see Theorem 1.4). By [15, Theorems 5 and 6], the minimum order of the canonical group of a spherical Eulerian triangulation of order n is (n - 2)/2. The computer program plantri [5] can be used to generate all spherical Eulerian triangulations of order up to 20. None of these triangulations have Z3 © Z3 as their canonical group; hence this group is also an exception.3 □ 2.4 Questions We conclude with three questions for future consideration. The first two address the remaining cases to be considered in order to resolve Question 1. Question 2. Let p = 2 be a prime, n > 3 if p > 5 and n > 2 if p = 5; does there exist a spherical latin bitrade with canonical group is isomorphic to Z^? Question 3. Let p be aprime and let 2 < al, a2,..., ak .If n > 1 + 2 J2 k=l (a - 1), does 2 The families of indicated in Figures 7 and 8 generalise to give abelian sandpile groups isomorphic to Z6m+5 ® Z6m+5, for all m > 1 and Z3m+ © Z3m+i, for all m > 1, respectively. However, we do not require these more general results to prove Theorem 2.9. 3 This proof that Z3 © Z3 is an exception is due to an anonymous referee. 184 Ars Math. Contemp. 13 (2017) 107-123 there exist a spherical latin bitrade with canonical group is isomorphic to zr; © ? Our final question arises naturally in response to the non-existence result Theorem 1.4. For a separated, connected latin bitrade (A, B) of genus greater than zero, the group AW is isomorphic to Z © Z ©C, but the minimal abelian representation (if one exists) is now a quotient of C, [1, Theorem 6]. Hence, we ask the following. Question 4. Does there exist a family of separated, connected latin bitrades for which the minimum abelian representation of one (or both) of the partial latin squares is isomorphic to Zk for arbitrary k? If so does such a family exist for a fixed genus? Acknowledgements The authors express their thanks to an anonymous referee for the argument showing that no spherical latin bitrade has a canonical group isomorphic to Z3 © Z3 as well as for their helpful comments and suggestions. The authors also express their thanks to the London Mathematical Society for a grant which enabled this research to by undertaken. References [1] S.R. Blackburn and T.A. McCourt, Triangulations of the sphere, bitrades and abelian groups, Combinatorica 34 (2014), 527-546, doi: 10.1007/s00493-014-2924-7. [2] C.P. Bonnington, M. Conder, M. Morton and P. McKenna, Embedding digraphs on orientable surfaces, J. Combin. Theory Ser. 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ARS MATHEMATICA CONTEMPORANEA 13 (2017) 187-206 Pursuit-evasion in a two-dimensional domain Andrew Beveridge Department of Mathematics, Statistics and Computer Science, Macalester College, Saint Paul, MN 55105 Yiqing Cai The Institute for Mathematics and Its Applications, University ofMinnesota, Minneapolis, MN 55455 Received 16 March 2016, accepted 3 September 2016, published online 3 March 2017 In a pursuit-evasion game, a team of pursuers attempt to capture an evader. The players alternate turns, move with equal speed, and have full information about the state of the game. We consider the most restrictive capture condition: a pursuer must become colocated with the evader to win the game. We prove two general results about this adversarial motion planning problem in geometric spaces. First, we show that one pursuer has a winning strategy in any compact cat(0) space. This complements a result of Alexander, Bishop and Ghrist, who provide a winning strategy for a game with positive capture radius. Second, we consider the game played in a compact domain in Euclidean two-space with piecewise analytic boundary and arbitrary Euler characteristic. We show that three pursuers always have a winning strategy by extending recent work of Bhadauria, Klein, Isler and Suri from polygonal environments to our more general setting. Keywords: Pursuit-evasion, lion and man, CAT(0) space, motion planning. Math. Subj. Class.: 91A24, 49N75, 53A04 1 Introduction A pursuit-evasion game in a domain D is played between a team of pursuers pi, p2,..., pk and an evader e. The pursuers win if some pi becomes colocated with the evader after a finite number of turns, meaning that the distance d(pi, e) = 0. When this occurs, we say that pi captures e. We consider the discrete time version of the game, which proceeds in turns. Initially, the pursuers choose their positions pi,p2,... ,p0k, and then the evader chooses his initial position e0. In turn t > 1, each pursuer pi moves from her current E-mail addresses: abeverid@macalester.edu (Andrew Beveridge), yiqingcai@ima.umn.edu (Yiqing Cai) Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 188 Ars Math. Contemp. 13 (2017) 107-123 position pt-1 to a point pt e B(pt-1, 1) = {x G D | d(pt-1,x) < 1}. If the evader has been captured, then the game ends with the pursuers victorious. Otherwise, the evader moves from et-1 to a point et e B(et-1,1). The evader wins if he remains uncaptured forever. We consider the full-information (full-visibility) game in which each player knows the environment and the locations of all the other players. Furthermore, the pursuers may coordinate their movements. Turn-based pursuit games in simply connected domains have been well-characterized: one pursuer is sufficient to capture the evader. Winning pursuer strategies have been found for environments in En [21, 16], and in simply connected polygons [14]. Taking a geometric viewpoint and using the weaker capture criterion d(p, e) < e for some constant e > 0, Alexander, Bishop and Ghrist [3] proved that a single pursuer can capture an evader in any compact cat(0) by heading directly towards the evader at maximum speed. We provide an alternate strategy for a compact cat(0) space that achieves d(p, e) = 0 in a finite number of turns. Our winning pursuer strategy is a generalization of lion's strategy, which has been used successfully in En [21] and in simple polygons [14]. We defer the description of this strategy to Section 2. Theorem 1.1. A pursuer p using lion's strategy in a compact cat(0) space D, captures the evader e by achieving d(p, e) = 0 after at most diam(D)2 turns. Theorem 1.1 implies that a single pursuer can become colocated with an evader in a simply connected, compact domain D c E2. In particular, this result holds for the polygonal setting in [14]. Notably, the general cat(0) viewpoint leads to an improved capture time bound for polygons, compared to the O(n ■ diam(D)) result in [14], where n is the number of vertices of polygon D. It is easy to construct compact domains that are evader win: removing one large open set from the middle of a simply connected domain tips the game in the evader's favor. Indeed, the evader can keep this large obstruction between himself and the pursuer, indefinitely. Such an open set is called an obstacle or hole in the environment. It is not hard to show that adding a second pursuer to this two-dimensional domain gives the game back to the pursuers. Adding multiple obstacles creates a distinct topology, and it is natural to wonder how many pursuers are needed to capture an evader in such an environment. The analogous question has been resolved for pursuit-evasion games in certain two-dimensional environments. Aigner and Fromme [1] proved that three pursuers are sufficient for pursuit-evasion on a planar graph. Bhadauria, Klein, Isler and Suri [8] showed that the analogous result holds in a two-dimensional polygonal environment with polygonal holes. We generalize the latter three-pursuer result to a broader class of geometric spaces. Our pursuit game takes place in a compact and path-connected domain D c E2. The set D contains a finite set of disjoint open obstacles O = {O1,O2,..., Ok}. The domain boundary is dD = {dO0, dO1, dO2,..., 3Ok} where we define dO0 to be the outer boundary of D, for convenience. We place two conditions on the boundary. First, dD can be decomposed into a finite number of analytic curves y(t) = (x(t), y(t)) for 0 < t < 1, where each of x(t),y(t) can locally be expanded as convergent power series. Second, we require that dD is a 1-manifold: for any x e dD, there exists an e > 0, such that B(x, e) n dD is homeomorphic to E1. In other words, we forbid self-intersections. For brevity, we say that a domain D satisfying these properties is piecewise analytic. We list three consequences of these conditions. First, the number of singular points on the boundary is finite. Second, the absolute value of the curvature at the nonsingular points of dD A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 189 is bounded above by some constant Kmax > 0. Third, there is a minimum separation dmin > 0 between boundary components: d(Oi,Oj) > dmin for all 0 < i < j < k. During the game, the pursuers will guard a sequence of geodesics; crucially, we will see in Section 4 that each of these geodesics is also piecewise analytic. This brings us to our main result. Theorem 1.2. Three pursuers can capture an evader in a compact domain in E2 with piecewise analytic boundary. The number of turns required to capture the evader for a domain with k obstacles is O(2k ■ diam(D) + diam(D)2). At a high level, our winning three-pursuer strategy builds on those found in [1, 8], and we are indebted to those previous papers. However, our geometric and topological approach is entirely new. In particular, our arguments are grounded in a careful investigation of the convexity, curvature and homotopy classes of geodesic curves in our domain. Furthermore, Theorem 1.2 significantly extends the class of known three-pursuer-win domains. Finally, we recently became aware of an unpublished technical report of Zhou et. al [22] that proves a similar result. Like our proof, their strategy adapts that of [8] to a more general setting. However, our underlying methodology is quite distinct: we use a homotopy based argument, while they use a geometric one. Furthermore, we devote more attention to the boundary of our region. In particular, we use analytic curves (rather than smooth curves) to avoid potential pathologies of geodesics. We provide more detail on finding guardable paths: we explain how to restrict ourselves to finding geodesics in closed sets, rather than in sets that are neither open nor closed (see Lemma 3.6 below). Finally, they use an endgame that requires two aggressive pursuers. We stick with lion's strategy due to its broad applicability to cat(0) spaces. 1.1 Related Work Pursuit-evasion games are a class of adversarial motion planning problems. Chung, Hollinger and Isler [11] provide an informative survey of pursuit games in mobile robotics. The interdisciplinary literature on pursuit-evasion games spans a range of settings and variations. Pursuit games have been studied in many environments, including graphs, in polygonal environments and in geometric spaces. Researchers have considered motion constraints such as speed differentials between the players, constraints on acceleration, and energy budgets. As for sensing models, the players may have full information about the positions of other players, or they may have incomplete or imperfect information. Typically, the capture condition requires achieving colocation, a proximity threshold, or sensory visibility (such as a non-obstructed view of the evader). Pursuit games enjoy a wide range of applications, including intruder neutralization, search-and-rescue, and environmental monitoring of tagged wildlife. In these settings, modeling an adversarial evader gives worst-case feasibility and time bounds. For an overview of pursuit-evasion on graphs, see the monograph by Bonato and Nowakowski [9]. Kopparty and Ravishankar [16] give a nice an introduction to pursuit in the polygonal setting. Research on pursuit-evasion spans nearly a century. In the 1930s, Rado posed the Lion and Man game in which a lion hunts the man in a circular arena. The players move with equal speeds, and the lion wins if it achieves colocation. At first blush, it seems that lion should be able to capture man, regardless of the man's evasive strategy. However, 190 Ars Math. Contemp. 13 (2017) 107-123 Besicovitch showed that when the game is played in continuous time, the man can follow a spiraling path so that lion can get arbitrarily close, but cannot achieve colocation [17]. However, when lion and man move in discrete time steps, our intuition prevails: lion does have a winning strategy [21]. The past decade has witnessed a renaissance of pursuit-evasion results in multiple disciplines. Prominent research efforts come from the robotics community, where pursuitevasion in polygonal environments is a productive setting for exploring autonomous agents. Pursuit-evasion has also thrived in the graph theory community, where it is known as the game of Cops and Robbers. More recently, researchers have started exploring pursuitevasion games in topological spaces. This is a natural evolution for the study of pursuitevasion games. Indeed, determining the number of pursuers required to capture an evader in a given environment becomes a question about its topology since the various loops and holes of the environment provide escape routes for the evader. The classic paper of Aigner and Fromme [1] initiated the study of multiple pursuers versus a single evader on a graph. In this turn-based game, agents can move to adjacent vertices, and the cops win if one of them becomes colocated with the robber. This paper introduced the cop number of a graph, which is the minimum number of pursuers (cops) needed to catch the evader (robber). Aigner and Fromme proved that the cop number of any planar graph is at most 3. This bound is tight, as the dodecahedron graph requires three cops. At a high level, their winning pursuer strategy proceeds as follows. Two cops guard distinct (u, v)-paths where u, v are vertices of the graph G. This restricts the robber movement to a subgraph of G. The third pursuer then guards another (u, v)-path, chosen so that (1) the robber's movement is further restricted, and (2) one of the other cops no longer needs to guard its path. This frees up that cop to continue the pursuit. This process repeats until the evader is caught. More recently, an analogous result was proven by Bhadauria, Klein, Isler and Suri [8] for pursuit-evasion games in a two-dimensional polygonal environment with polygonal holes. In this turn-based game, an agent can move to any point within unit distance of its current location. Like Aigner and Fromme, they use colocation as their capture criterion. Bhadauria et al. prove that three pursuers are sufficient for pursuit-evasion in this setting, and that this bound is tight. The pursuer strategy is inspired by the Aigner and Fromme strategy for planar graphs: two pursuers guard paths that confine the evader while the third pursuer takes control of another path that further restricts the evader's movement. Of course, the details of the pursuit and the technical proofs are quite different from the graph case. Their proofs make heavy use of the polygonal nature of the environment, both to find the paths to guard and to guarantee that their pursuit finishes in finite time. Just as the proofs of Bhadauria et al. were inspired by Aigner and Fromme, our proof of Theorem 1.2 is inspired by those for the polygonal environment. Bhadauria et al. actually give two different winning strategies for three pursuers. At a high level, these strategies progress in the same way, but the tactics for choosing paths and how to guard them are different. Herein, we adapt their shortest path strategy to our setting. Our more general geometric environment introduces a distinctive set of challenges to overcome. In particular, we do not have a finite set of polygonal vertices to use as a backbone for our guarded paths. Instead, we rely on homotopy classes to differentiate between paths to guard. Looking beyond the high-level structure of our pursuer strategy, the arguments (and their technical details) in this paper are wholly distinct from those found in [8], and our result applies to a much broader class of environments. A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 191 Finally, we note that our result follows in the footsteps of other recent explorations of pursuit-evasion games in general geometric and topological domains. Pursuit-evasion games in such spaces have further applications in robotics, where agents must navigate and coordinate in high dimensional configuration spaces. Alexander, Bishop and Ghrist helped to pioneer this subject, studying pursuit-evasion games with the capture condition e) < e for some constant e > 0 (rather that colocation). In [3], Alexander, Bishop and Ghrist prove that a single pursuer can capture an evader in any compact cat(0) space: the simple pursuit strategy of heading directly towards the evader is a winning strategy. In [5], these authors explore the simple pursuit strategy in unbounded cat(K) spaces with positive curvature K > 0, developing connections between evader-win environments and the total curvature of the pursuer's trajectory. Finally, in [4], they consider pursuit games in unbounded Euclidean domains using multiple pursuers. They provide conditions on the initial configuration of the players that guarantee capture, generalizing (and amending) results of Sgall [21] and Kopparty and Ravishankar [16]. 1.2 Preliminaries We introduce some notation and review some concepts and results from algebraic topology [13]. We then prove three lemmas about convex paths in two-dimensional compact regions with piecewise analytic boundary. A topological space is a set X along with a collection of subsets of X, called open sets that satisfy a sequence of axioms [7]. A map f : X ^ Y between two spaces is continuous when the inverse image of every open set in Y is open in X. A path n : [0,1] ^ D is a continuous map from the interval [0,1] to D, with initial point n(0) and terminal point n(1). The length l(n) of this path is its arc length in Euclidean space E2. A path n is a loop when n(0) = n(1). A simple path has no self-intersections, meaning that n is injective. By abusing notation, we write x e n when x = n(t) for some t e [0,1]. For x, y G n, we use n(x, y) to denote the subpath connecting these points. The space X is path-connected if there exists a path between any pair of points x, y e X. A homotopy of paths is a family of maps ft : [0,1] ^ X, t e [0,1], such that the associated map F : [0,1] x [0,1] ^ X given by F(s, t) = ft(s) is continuous, and the endpoints ft(0) = x0 and ft(1) = xi are independent of t. The paths f0 and f are called homotopic. The relation of homotopy on paths with fixed endpoints is an equivalence relation and we use [f ] to denote the homotopy class of the curve f under this relation. The set [f] of loops in X at the basepoint x0 forms a group under path composition, called the fundamental group of X at the basepoint x0. The space X is simply connected when it is path-connected and its fundamental group is trivial. For example, a subspace X of E2 is simply connected if and only if it has the same homotopy type as a 2-disc. We now turn to some geometric properties of paths in E2. The distance d(x, y) between points x, y e X is the length of a shortest (x, y)-path in X. When restricting ourselves to R C X, we use dR (x, y) to denote the distance between these points in the subdomain. We will frequently consider a subdomain R enclosed by two simple (u, v)-paths n1, n2. We denote such a set as R[n1, n2] C X. Fora C2 path 7 : [0,T] ^ E2, its curvature at Y(t) is defined as «(t) = ± IIy'^^'Wl^11, with the sign positive if the tangent turns counterclockwise, and negative if the tangent turns clockwise. The smoothness of a piecewise analytic curve 7 : [0,1] ^ E2 ensures that its absolute curvature is bounded at its nonsingular points. If 7 is piecewise C2 and 192 Ars Math. Contemp. 13 (2017) 107-123 u u (a) (b) Figure 1: (a) A piecewise convex (u, v)-path n. (b) Shortcutting a non-convex (u, v)-path ni. continuous, with t0 < t1 < • • • < tn as the preimages of the singular points, then its total curvature is where Qi is the exterior angle at 7^), and Qn = 0 when 7(t0) = 7(tn). This brings us to the celebrated Gauss-Bonnet Theorem which relates the total curvature of a closed curve with the Euler characteristic of its enclosed region. In our setting, the Euler characteristic equals 1 - k, where k is the number of obstacles in the region R. Theorem 1.3. [Gauss-Bonnet Theorem, cf. [12]] Given a compact region R C E2 with boundary dR, we have «total (dR) =2nx(R), where x(R) is the Euler characteristic of R. We use the Gauss-Bonnet Theorem to understand the effect of obstacles on shortest paths. In particular, we will consider pairs of paths n1, n2 with shared endpoints u, v. These paths will be piecewise analytic (so that they have a finite number of singular points). Our goal is to prove that if the shortest (u, v)-path is not unique, then each shortest path must touch an obstacle in the given region. We begin with a definition of convexity, which we define for the broader family of piecewise C2 smooth curves; an example is shown in Figure 1(a). Definition 1.4. Let n : [0,1] ^ E2 be a piecewise C2 smooth curve in E2. Then n is convex when the following holds for any point x e n\{u, v}: (a) If n is C2 smooth at x, then the curvature at x is nonpositive; (b) If n is not C2 smooth at x, then the tangent line at x turns clockwise by an angle 0 < Q < n. The definition for a concave curve is similar, but the curvature must be nonnegative and the tangent line must turn counterclockwise. Note that if the (u, v)-curve n is convex, then the (v, u)-curve r given by r(t) = n(1 -1) is concave. Lemma 1.5. Let n1 and n2 be two piecewise analytic (u, v)-paths with n1 n n2 = {u, v} and such that R[n1, n2] lies to the left of n1. If the curve n1 is a shortest (u, v)-path in R[n1, n2] and touches no obstacle inside R[n1, n2], then n1 is convex in R[n1, n2]. A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 193 Note that the curve ni does not to need to be a straight line: see Figure 1(a) for an example. Similar convex bounding curves will arise as the pursuers remove obstacles from the evader's reach. Proof. We prove the lemma by contradiction. Suppose that there exists an x G n\{u, v} where the convexity of n1 in R[n1, n2] is violated. Either (a) n1 is C2 smooth at x, but the curvature at x is positive, or (b) n1 is not C2 smooth at x, and the tangent line turns counterclockwise by an angle 0 < 0 < n at x, creating a non-convex corner. Let d0 denote the minimum distance between n1 and any obstacle O G R[n1, n2] with n1 n dO = 0. Suppose that the curvature at x is positive, see Figure 1(b). There must be a C2 subpath nx between y 1 and y2 of n1 around x with positive curvature. Using the lower bound d0 on the separation between n1 and any obstacles inside R[n1, n2] and n2, we may take y1, y2 to be close enough so that the line segment A connecting y1 and y2 lies inside R[n1, n2] and does not encounter any obstacles. Replacing nx with A creates a path that is strictly shorter than n1, contradicting the minimality of n1. Next suppose there is a non-convex corner at x. By an analogous argument to the previous case, we can create a short-cut A around x to make a shorter path than n1, a contradiction. □ Lemma 1.6. Let n1, n2 be two (u, v)-paths with n1 n n2 = {u, v}. Suppose that n1 is a convex and piecewise analytic (u, v)-path in R[n1, n2], and let n2 be a convex and piece-wise analytic (v, u)-path in R[n1, n2]. Then n1 and n2 are both straight lines connecting u, v. Proof. Let Q = Q[n1, n2] be the simply connected, closed region between n1 and n2 (so we ignore all obstacles in D). The concatenation of n1(u, v) and n2(v, u) is a loop dQ bounding the simply connected region Q. By the Gauss-Bonnet Theorem 1.3, the total curvature along dQ equals 2nx(Q) = 2n. We decompose the total curvature of dQ as the sum of total curvature of n1 and n2 respectively, and the exterior angles at u and v. Because of convexity, both n1 and n2 have total curvature no greater than 0. As for the two angles at u, v, neither can exceed n. Therefore the total curvature of the loop does not exceed 2n, and could only achieve 2n when «total(n1) = Ktotal(n2) = 0. Therefore, n1 and n2 are both straight lines connecting u and v. □ Lemma 1.7. Suppose n1, n2 are two shortest (u, v)-paths in the region R = R[n1, n2], with n1 n n2 = {u, v}. Then each of n1, n2 touches at least one obstacle in R. Proof. Suppose that the conclusion is false. Without loss of generality, n1 does not touch any obstacles in R. By Lemma 1.5, the (u, v)-path n1 is convex in R. Let Q be the simply connected region obtained by removing the obstacles in R. We have 1(n1) = 1(n2), so they are both shortest (u, v)-paths in the simply connected environment Q. Therefore n2 is also convex by Lemma 1.5, if parameterized as a path from v to u. By Lemma 1.6, n1 and n2 are both straight lines connecting u, v, which contradicts n1 n n2 = {u, v}. This proves that when there is more than one shortest (u, v)-path, each of these paths must touch an obstacle inside R. □ This concludes our topological and geometric preliminaries. 194 Ars Math. Contemp. 13 (2017) 107-123 c •' P e 1 Figure 2: Lion's strategy in E2. On each move, the pursuer moves on the line segment connecting the center c to the evader, and increases her distance from c. 2 Lion's Strategy in a cat(0) space In this section, we describe a winning strategy for a single pursuer in a compact cat(0) domain, and prove Theorem 1.1. Our strategy generalizes lion's strategy for pursuit in E2, introduced by Sgall [21]. This strategy was adapted for pursuit in polygonal regions by Isler, Kannan and Khanna [14]. Their adaptation relies heavily on the vertices of the polygon P and gives a capture time of n • diam(P)2, where n is the number of vertices of P. We give a version of lion's strategy that succeeds in any compact CAT(0) domain D (including polygons) with capture time bounded by diam(D)2. Sgall's version of lion's strategy proceeds as follows. Fix a point c as the center of our pursuit, see Figure 2. The pursuer starts at p = c and the evader starts at some point e. On her first move, the pursuer moves directly towards e along the line ce. Considering a general round, the pursuer will be on the line segment between c and e prior to the evader move. After the evader moves to e' G B(e, 1), the pursuer looks at the circle centered at p with radius e. If e is inside this circle, then the pursuer can capture the evader. Otherwise, this circle intersects the line segment ce' at two points. The pursuer moves to the point p' that is closer to e. Lemma 2.1 (Sgall [21]). A pursuer using lion's strategy in E2 re-establishes her location on the line segment between c and the evader. Furthermore, if the evader moves from e to e' andthe pursuer moves from p to p' then d(c,p')2 > d(c,p)2 + 1. Before generalizing lion's strategy, we introduce of the basics of a cat(0) geometry; see [10] for a thorough treatment. A complete metric space (X, d) is a geodesic space when there is a unique path n(x, y) whose length is the metric distance d(x,y). This path n(x, y) is called a geodesic (or shortest path). A triangle Axyz between three points x, y, z G X is the triple of geodesics n(x, y), n(y, z), n(z, x). To each Axyz G X, we associate a comparison triangle Axyz c E2 whose side lengths in E2 are the same as the lengths of the corresponding geodesics in X. The complete geodesic metric space (X, d) is cat(0) when no triangle in X has a geodesic chord that is longer than the corresponding chord in the comparison triangle. In other words, pick any triangle Axyz and any points u G n(x, y) and v G n(y, z). Let U G xy and V G yz be the corresponding points, chosen distancewise on the edges xy and yz. If the space X is cat(0) then dx (u, v) < dg2 (U, V). Colloquially, this is called the "no fat triangles" property, since it also implies that the sum of the angles of the triangle is not greater than n. Our cat(0) lion's strategy generalizes the extended lion's strategy for polygons of Isler et al. [14]. The pursuer starts at a fixed center point c and her goal is to stay on the shortest path n(c, e) at all times. In particular, assume that pt is on the shortest path n(c, et) and A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 195 that the evader moves from et to et+i. If d(pt, et+i) < 1 then the purser responds by capturing the evader. Otherwise, the pursuer draws the unit circle C centered at pt and moves to the point in C n n(c, et+1) that is closest to et+1. Lemma 2.2 (Lion's Strategy). A pursuer using lion's strategy in a cat(0) space (X, d) reestablishes her location on the line segment between c and the evader. Furthermore, if the evader moves from e to e' and the pursuer moves from p to p' then d(c,p')2 > d(c,p)2 + 1. Proof. Suppose that p G n(c, e) and then the evader moves to e'. Consider the cat(0) triangle Acee' and its comparison triangle Acee' in E2. Look at the corresponding E2 pursuit-evasion game with the pursuer at p G ce. By Lemma 2.1, the pursuer can move to apointp' G ce' such that dE2(c,pC')2 > dE2(c,pe)2 + 1. Since there are no fat triangles in X, we have dX(p, p') < dE2 (p,p') where p' G n(c, e') is the point corresponding to p'. Therefore, in our original game, the pursuer can move to the point p' G n(c, e'), which satisfies dX(c,p')2 > dX(c,p)2 + 1. □ Finally, we prove Theorem 1.1: lion's strategy succeeds in a cat(0) domain. Proof of Theorem 1.1. Consider a pursuit-evasion game in the compact cat(0) domain D. Pick any c G dD as our center point. Using lion's strategy, the pursuer increases her distance from c with every step by Lemma 2.2, so she captures the evader after at most diam(D)2 rounds. □ 3 Minimal Paths and Guarding The key to our pursuit strategy is the ability of one pursuer to guard a shortest path, meaning that the evader cannot cross this path without being caught by a pursuer. When this shortest path splits the domain into two subdomains, the evader will be trapped in a smaller region. We refer to this region as the evader territory. In fact, we will be able to also guard a "second shortest path" when the shortest path is already guarded. The definitions and lemmas in this section are adaptations of the minimal path formulations introduced in [8] and further developed in [6]. Recall that we use d(x, y) to denote the length of a shortest (x, y)-path in D. In addition, we will use X and X to denote the interior and the closure of a set X, respectively. Definition 3.1. Let X c D be a path-connected region. The simple path n c X is minimal in X when for any y 1, y2 G n and any z G X, we have dn (y1, y2) < dX (y1, z) + dx (z,y2). Definition 3.2. Let Z c X and let n be a minimal (u, v)-path in Z where u, v G dZ. Then the path projection with anchor u is the function n : Z ^ n defined as follows. If dZ(u, z) < dZ(u, v) = dn(u, v), then n(z) is the unique point x G n with dn(u,x) = dZ(u, z). For the remaining z G Z\n, we set n(z) = v. We make a few observations. If X = D, then a shortest (x, y)-path is always a minimal path in D. In this case, we can define the path projection n : D ^ n. When X C D, we might have dX (x1, x2) > d(x1, x2). In this case, a shortest path in X will be minimal in X, but it will not be minimal in D. Next, we show that a path projection is non-expansive, meaning that distances cannot increase. 196 Ars Math. Contemp. 13 (2017) 107-123 Lemma 3.3. Let n : Z ^ n be a path projection onto a minimal path in Z. Then dn(n(zi),n(z2)) < dz(^1,^2) for all zi,z2 € Z. Proof. The proof is a straight-forward argument using the triangle inequality. We consider the case z1, z2 € Z with d(u, z1) < d(u, z2) < d(u, v). We have dz(z2,zi) > dz(z2,u) - dz(zi,u) = dn(n(z2),u) - dn(n(zi),u) = dn(n(z2 ),n(zi)). The other non-trivial cases are argued similarly. □ A single pursuer can turn a minimal path n into an impassable boundary: once the pursuer has attained the position p = n(e), the evader cannot cross n without being captured in response. The proof of the following lemma is similar to the analogous result in [6], but we include this brief argument for completeness. Lemma 3.4 (Guarding Lemma). Let n : X ^ n be a path projection onto the minimal (u, v)-path n c X. Consider a pursuit-evasion game between pursuer p and evader e in the environment X. (a) After O(diam(X)) turns, the pursuer can attain pl = n(et-1). (b) Thereafter, the pursuer can re-establish ps+1 = n(es) for all s > t. (c) If the evader moves so that a shortest path from es-1 to es intersects n, then the pursuer can capture the evader at time s + 1. Proof. To achieve (a), the pursuer moves as follows. First, p travels to u, reaching this point in O(diam(X)) turns. Next, p traverses along n until first achieving d(u,pi) < d(u, n(ei_i)) < d(u,pi) + 1. If pi = n(ei_i) then we are done. Otherwise, when the evader moves, we either have d(u,pi) - 1 < d(u,n(ei)) < d(u,pi) + 1 or d(u,pi) + 1 < d(u, n(ei)) < d(u,pi) + 2 by Lemma 3.3. In the former case, p can move to n(ei) in response, achieving her goal. In the latter case, p will increase her distance from u by one unit, re-establishing d(w,pi+i) < d(u,n(ei)) < d(w,pi+i) + 1. This latter evader move can only be made O(diam(X)) times, after which the pursuer acheives p = n(e). Next, suppose thatps = n(es_i) and that es_i € X\n. The pursuer can stay on the evader projection by induction since dn(ps,n(es)) = d^(^(es-1), n(es)) < d(es-i,es) < 1, so (b) holds. As for (c), suppose that a shortest path from es_i to es includes the point y € n. Then d(ps, es) < d^(^(es-1), y) + d(y, es) < d(es-1, y) + d(y, es) = d(es-1, es) = 1. Therefore the pursuer can capture the evader on her next move. □ The Guarding Lemma is the cornerstone of our pursuer strategy. When the pursuer moves as specified in the lemma, we say that she guards the path n. In Section 4, our pursuers will repeatedly guard paths chosen to reduce the number of obstacles in the evader territory. Once the evader is trapped in a region that is obstacle-free, we have reached the endgame of the pursuit. A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 197 Lemma 3.5. Suppose that the evader is located in a simply connected region R whose boundary consists of subcurves of the original boundary dD and two guarded paths ni and n2. If the evader remains in R, then the third pursuer can capture him infinite time. If the evader tries to leave the region through n1 or n2, then he will be captured by the guarding pursuer. Proof. By Lemma 3.4, if the evader tries to leave this region, he will be caught by either p1 or p2. If the evader remains in this component, then Theorem 1.1 guarantees that pursuer p3 captures the evader in a finite number of moves. □ The remainder of this section is devoted to identifying guardable paths that touch obstacles. Guarding such a path will neutralize the threat posed by the obstacle. First, we consider the case when p1 guards the unique shortest (u, v)-path n1 that touches an obstacle O in the evader region. The objective of p2 is to guard another (u, v)-path n2 of a different homotopy type. This path can be guarded even when n2 is longer than n1, provided that any path shorter than n2 also intersects n1. Lemma 3.6. Suppose that the evader territory R = R[n1, A] is bounded by the unique (u,v)-shortest path n1 and another boundary curve A. Furthermore, suppose that n1 touches an obstacle O and that n1 is guarded by p1. Then we can find a (u, v)-path n2 c R with the following properties: (a) O c R[n1, n2], so that the homotopy type of n2 is different than that of n1; and (b) n2 is guardable by p2, provided that n1 remains guarded by p1. A naive attempt to find such a path is to pick some x e n1 n O and find a shortest path that does not include the point x. However, R\{x} is not a closed set, which would complicate our argument. Furthermore, it could be that the next shortest path includes x without using this point as a shortcut around the obstacle O, as shown in Figure 3(b).1 We handle both problems by removing a small and well-chosen open region A near x, rather than removing the point x. The delicate choice of A relies on two consequences of our piecewise analytic boundary: the finite upper bound Kmax > 0 on the curvature and the minimum distance dm;n > 0 between boundary components. Proof. First, suppose that n1 n dO includes a continuous subcurve C c n1. Pick x e C and e > 0 so that B(x, e) n R c O. Let R' — R\B(x, e), which effectively absorbs the obstacle into the boundary, see Figure 3 (a). The region R' is closed, so there is a well-defined shortest (u, v)-path n2 c R'. The path n2 is guardable in R', and therefore it is guardable by p2 in R, provided that p1 guards n1. Indeed, any shorter path in R must go through the point x, so Lemma 3.4 guarantees that an evader using such a path will be caught by p1. Finally, we note that n1, n2 have distinct homotopy types because o c Rp1, n2]. Next, we consider the case where n1 n O contains no continuous curves: we just focus on the first point x e n1 n O encountered as we move from u to v. Locally around x, the path n1 and the boundary dO separate R into two external regions (outside of n and inside dO) and two internal regions, see Figure 3 (b). The shortest path n1 does not self-intersect, so locally near x, this path consists of two curves meeting at x, creating an !We note that this unusual circumstance is overlooked in [8], where it can occur during their minimal path strategy. This case can be easily handled in a manner analogous to our approach, but based on the visibility graph of their environment. 198 Ars Math. Contemp. 13 (2017) 187-206 O ni v (a) A A n^ ¿p (b) (c) A vA (d) Figure 3: Finding the second shortest path. (a) When n1 n dO contains a curve, we can remove a small open ball. (b) The shortest path n1 touches obstacle O at x. The second shortest path n2 goes around O, but includes the point x. (c) Finding n2 requires removing a small, open, triangular set A between n1 and O, and then finding the shortest (u, v)-path in the closed set R\A. (d) Any path that crosses the line segment yz can be short-cut. x u u v interior angle smaller than 2n. Therefore, at least one of the two interior angles made by n1 and the obstacle tangent line(s) at x is strictly less than n. This local region is where we will remove our triangular open set. Without loss of generality, suppose that the subpath n1(x, v) helps to bound this local region. Take points y G n1(x, v) and z G dO (traveling counterclockwise from x) such that 0 < dni (x, y) = ddO (x, z) < dmin/2, and the angle Zyxz < n. Let A c B(x, dmin) be the closed region with endpoints (x, y, z), where the third curve is the unique shortest (y, z)-path r, see Figure 3 (c). The bound on the absolute curvature Kmax allows us to choose our y, z so that the region A is essentially triangular. Since dmin is the minimum distance between obstacles, A is obstacle-free, so r is a straight line segment. We remove the relatively open set A' = A\r from our domain. We then find the shortest (u, v)-path n2 in the closed set R = R\A'. We claim that p2 can guard n2 in R, provided that p1 guards n1. As in the previous case, the shorter paths that go through x are not available to the evader. Therefore, we must show that any path in R that visits A' is longer than n2. Such a path A must enter and leave A' through r, say at points a, b, see Figure 3 (d). However, the subpath A(a, b) can be replaced with the unique shortest path r(a, b) without changing the homotopy type, a contradiction. Once again, n1, n2 have distinct homotopy types because O c R[n1, n2]. □ We refer to the paths n1, n2 from Lemma 3.6 as a guardable pair. Provided that the shortest (u, v)-path n1 is guarded, the "second shortest (u, v)-path" n2 can also be guarded. The following corollary is a variation of the lemma. Corollary 3.7. Let n1, n2 be (u, v)-paths that are guarded by p1,p2, respectively. Suppose that for i = 1, 2, the path n touches an obstacle Oit where O1 = O2. Then we can find a path n3 with the following properties: (a) the homotopy type of n3 is different than the homotopy types of n1 and n2, and in particular, Oj G Rpj, n3] for i = 1,2; and (b) n3 is guardable by p3, provided that n1, n2 remain guarded by p1, p2. Proof. The proof is similar to the proof of Lemma 3.6. This time, we must remove an open set A near x g n1 n O1 and an open set B' near y G n2 n O2. We then find n3 in A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 199 R\(A' U B'). □ This concludes our search for guardable paths that touch obstacles. The next section lays out the three-pursuer strategy for capturing the evader in a two-dimensional domain. 4 Shortest Path Strategy In this section, we prove Theorem 1.2: three pursuers can capture an evader in a two-dimensional compact domain with piecewise analytic boundary. We adapt the the shortest path strategy of Bhaudaria et al. [8] to our more general setting. In particular, our guard-able path lemmas from Section 3 supplant their use of polygon vertices to find successive paths. Their algorithm guarantees success by reducing the number of polygon vertices in the evader territory. Instead, we keep track of the threat level of obstacles to argue that the evader becomes trapped in a simply connected region. Our pursuit proceeds in rounds. At the start of a round, at most two pursuers guard paths. The third pursuer moves to guard another path with the goal of eliminating obstacles from the evader territory. This third path will either be a shortest path, or it will create a guardable pair with the currently guarded path(s). Once this third path is guarded, the evader is trapped in a smaller region, which releases one of the other pursuers to continue the process. This continues until the evader is trapped in a simply connected region, where the free pursuer can capture the evader by Lemma 3.5. We start by showing that the boundary of the evader territory is always piecewise analytic, after recalling two definitions. First, the endpoints of a line segment touching the boundary dV are called switch points. Second, a point x is an accumulation point (or limit point) of a set S when any open set containing x contains an infinite number of elements in S. We make use of the following result about the interaction of a geodesic with the boundary of the domain. Theorem 4.1 (Albrecht and Berg [2]). If M is a 2-dimensional analytic manifold with boundary embedded in E2, and y is a geodesic in M, then the switch points on y have no accumulation points. We restrict ourselves to analytic boundary, instead of smooth (C2, or even Cbound-ary, to avoid some potentially pathological behavior of geodesics. For example, Albrecht and Berg [2] construct a geodesic in Cenvironment, that achieves a Cantor set of positive measure as the accumulation of switch points. This unusual geometry hampers our ability to confine the evader in a well-defined connected component. Theorem 4.1 ensures that our new evader territory will be bounded by piecewise analytic curves. Lemma 4.2. Let V be a compact domain with piecewise analytic boundary. If n is a shortest path in V, then n is piecewise analytic. Furthermore, if V\n is disconnected, then it contains finitely many connected components, and the boundary of each connected component is piecewise analytic. Proof. Let B c d(n\dV) be the set of switch points. We claim that B is finite. Otherwise, there must be an accumulation point of B since l(n) is finite, contradicting Theorem 4.1. Now we can use the finite set B as endpoints to partition n so that each subcurve is either in the boundary dV, or in the interior V. Since any shortest path in the interior V must be a line segment, the path n is piecewise analytic. For each connected component of V\n, its boundary is a subset of n U dV, hence is piecewise analytic. □ 200 Ars Math. Contemp. 13 (2017) 107-123 In order to prove Theorem 1.2, we will show that our pursuit succeeds in finite time. To aid in this effort, we assign a threat level to each of the k obstacles in the original domain. These threat levels will reliably decrease during pursuit. An obstacle is in one of three states: dangerous, safe, or removed. A removed obstacle lies outside the evader territory. A safe obstacle lies in the evader territory and touches a currently guarded path. This obstacle is not a threat because the evader cannot circle around the object without being captured. The remaining obstacles are dangerous. Finally, we say that the evader territory is dangerous if it contains at least one dangerous obstacle. At the start of pursuit, all obstacles are dangerous. So long as there are still dangerous obstacles, a round consists of taking control of a guardable path. This effort succeeds in a finite number of moves by Lemma 3.4. We will show that after at most two rounds, either a dangerous obstacle transitions to safe/removed, or a safe obstacle transitions to removed. This is our notion of progress: after at most 2k rounds, the evader territory is not dangerous. From here forward, we focus on the transition of the threat levels of obstacles. In general, our evader territory will be bounded by part of the domain boundary dD and by at most two guarded paths ni, n2. At the end of a round, the evader territory will be updated, bounded in part by updated paths ni, n2. If these guarded paths intersect or share subpaths, then the evader is actually trapped in a smaller region by Lemma 3.4. When this is the case, we advance the endpoint(s) of our paths so that these are the only point(s) shared by our paths. This obviates the need to discuss degenerate cases. The first round is an initialization round, so all obstacles might still be dangerous when this round completes. However, we will be able to neutralize at least one obstacle in the subsequent round. To kick off the first round, we pick points u, v e dD, chosen so that they divide the outer boundary into two curves A1, A2 of equal length, see Figure 4(a). Let n1 be a shortest (u, v)-path; if there are multiple shortest paths (in which case each touches an obstacle), then we pick one arbitrarily. Using Lemma 3.4, p1 moves to guard n1. The round ends when p1 has attained guarding position, trapping the evader in a subdomain that is bounded by n1 and one of A1, A2. The evader could be trapped in a smaller pocket region between n1 and a subcurve A3 of an obstacle O C D, see Figure 4(b). In the latter case, the obstacle O is marked as removed and we treat A3 as the new outer boundary. After updating the evader territory R, any obstacle O C R is marked as removed. Any obstacle O C R that touches n1 or n2 is marked as safe. For the remainder of the game, the evader territory is one of the following types. • Type 0 region: A region containing no dangerous obstacles. • Type 1 region: A dangerous three-sided region bounded by a (u, v)-shortest path n1, a (u, w)-shortest path n2 and a (v, w)-path A C dD. No obstacle touches both ni, n2. • Type 1' region: A dangerous two-sided region bounded by a (u, v)-shortest path n1 and a (u, v)-path A C dD. We treat this as a special case of the previous type, where n2 consists of the single point w = u. This point is on n1, so it is guarded by p1. • Type 2 region: A dangerous two-sided region bounded by (u, v)-paths n1, n2, each of which touches an obstacle in the evader territory. The path n1 is a shortest (u, v)-path in this region. The path n2 might also be a shortest (u, v)-path, or it could be a "second shortest path," meaning that it is a shortest (u, v)-path among the set of (u, v)-paths that are not homotopic to n1. No obstacle touches both n1, n2. A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 201 Ai Ai A2 (a) nr- I Q A3 A2 (b) Figure 4: The shortest (u, v)-path n i guarded in round one. (a) The path n i partitions the outer boundary to subcurves A i, A2 of equal length. (b) The evader may be trapped in a pocket between the path n i and the boundary subcurve A3 of obstacle O. u u v • Type 3 region: a dangerous 4-sided region bounded by a (u, v)-shortest path n i, a (w, x)-shortest path n2, a (v, w)-path A i from the boundary and a (u, x)-path A2 from the boundary. These vertices are arranged so that they are ordered clockwise as u, v, w, x. No obstacle touches both n i, n2. For example, after the initialization round, the evader territory is a type 1' region, bounded by a guarded path and part of the boundary dD. Finally, we emphasize that Lemma 4.2 ensures that the boundary of the evader region is always piecewise analytic, since it consists of sub-curves of the piecewise analytic boundary along with one or more shortest paths. We now describe the different types of rounds. In regions of type 1, 1' and 2, we will always transition at least one obstacle. At the end of such a round, the evader could now be trapped in a region of any type. Type 3 rounds are slightly different. Our primary goal is to trap the evader in a type 1 region, where we will surely make progress in the subsequent round. However, it is possible to transition an object via a type 3 move (just as in the initialization round). In this case, we make immediate progress, and the evader could then be trapped in a region of any type. First we consider type 1 regions. This also handles type 1' regions as a special case. We use the following lemma to identify a point x G A and a shortest (u, x)-path to guard during this round. Lemma 4.3. Let shortest paths n0(u, v), n i(u, w) and boundary path A(v, w) bound a type 1 environment R. If R contains obstacles, then there exists a point x G A such that there are multiple shortest (u, x)-paths in R each of which touches at least one obstacle. Proof. Parameterize the boundary path as A : [0,1] ^ R. We prove the lemma by contradiction. Suppose that for every t G [0,1], the shortest (u, A(t))-path nt is unique. Denote its length by 1(nt) = d(u, A(t)). Define the function n(t) to be the number of obstacles in the region Rt bounded by n0, nt and A. The function n(t) is well-defined by the uniqueness of each nt. Furthermore, n(0) = 0, and n(1) > 0, so there must be a jump discontinuity somewhere in [0,1]. Let s = inf{t G [0,1] : n(t) > 0}. Case 1: n(s) > 0 where 0 < s < 1. (Recall that n(0) = 0, so s > 0.) Let r be a shortest (u, A(s))-path that is of the same homotopy class as n0. The choice of s and 202 Ars Math. Contemp. 13 (2017) 107-123 u A v u \ / 1K T-r V \ o n2 / \ / \ y & t { x w v x (a) (b) u K u A / » no ni nil* ^2 u ,1 ^ Al' n^ \n2 w v x (c) w v = x w v x w (d) (e) 3 v Figure 5: Representative examples of a type 1 move, where we transition to (a) a type 1 region, (b) a type 1 or type 1' region, (c) a type 1 or a type 3 region, (d) a type 1 or type 2 region, (e) a type 1 or type 3 region. the uniqueness of ns guarantee that l(T) > /(ns). Also no obstacles are contained in the region bounded by n0, r and A(0, s). By the definition of r, for all t G [0, s), we have ¿(r) < ¿(rt) + ¿(A(t, s)). By the continuity of 1(nt) = d(u, A(t)) with respect to t g [0, s), we have /(r) < /(nt) + /(A(t, s)), so /(r) < lim (z(no + i(A(t, s))) = i(ns) + o = i(ns). t—s- This contradicts /(r) > /(ns), so ns is not the unique shortest (u, A(s))-path. Case 2: n(s) = 0, where 0 < s < 1. Let {s^} be an infinite sequence si ^ s+, such that n(sj) > 0 for all i. There are finite number of obstacles, so by taking a subsequence if necessary, we can assume that the shortest paths {ns.} are of the same homotopy class. Let r be the shortest (u, A(s))-path of this homotopy class. We have /(r) < /(nsi) + /(A(s, sj)) for all i, and therefore /(r) < lim (i(nsi) + i(A(s,sj))) = /(ns), i—^^o where the limit holds by the continuity of distances in the region. However, this contradicts the uniqueness of ns which would require /(ns) < /(r). Thus we can conclude that are multiple shortest (u, x)-paths. By Lemma 1.7, each of these paths touches at least one obstacle. □ Having found the next path to guard, we now prove that we transition an object during a type 1 move. Lemma 4.4. Suppose that the evader is trapped in a type 1 (or type 1') region. Then the third pursuer can guard a path that transitions an obstacle state. Proof. By Lemma 4.3, there is some point x G A with multiple shortest (u, x)-paths, each of which touches an obstacle. Let n3 be one of these shortest (u, x)-paths. If x = v then we take a path n3 = ni. Similarly if x = w we choose n3 = n2. When x G {v, w}, we can choose n3 arbitrarily from the collection of (u, x)-shortest paths. Pursuer p3 moves to guard n3, which traps the evader in either R[n;, n3] or R[n3, n2]. Any obstacles in the other region are marked as removed. Without loss of generality, let O c R[n;, n3] be an obstacle touched by n3, see Figure 5(a). Suppose that prior to p3 guarding n3, the object O was dangerous. If e G R[n3, n2] A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 203 then O transitions to removed. If e e R[n1, n3] then O transitions to safe. However, we may be in a more advantageous position, shown in Figure 5(b): the evader could be trapped in a pocket between obstacle O and path n3. In this case, the new evader territory is type 1' and the obstacle O is marked as removed, since it is now part of the outer boundary of the evader territory. Next, suppose that O was already safe, touched by n1. If e e R[n2, n3] after p3 guards n3, then O transitions from safe to removed. If the evader is trapped in a pocket region between O and n3, we proceed as in the previous case. Otherwise, we have e e R[n1, n3] and the obstacle O separates R[n1, n3] into disjoint regions, as shown in Figure 5(c). The evader is trapped in one of these two subregions because both n1, n3 are guarded. Let A' be the subcurve of dO that bounds the effective evader territory. We update the evader territory appropriately, bounded by A' and subpaths of n1, n3, and perhaps part of A. The result is a region of type 1 or 3. The obstacle O is marked as removed: it is now part of the boundary. This reduces the number of safe obstacles. When n3 touches multiple obstacles, each of them transitions to a lower threat level. Figures 5(d) and (e) show that we can also end up in a type 2 or 3 region, depending on the configuration of these obstacles and the location of the evader at the end of the round. □ Next, we consider a type 2 region. Such a region is bounded by (u, v)-paths n1, n2 that form a guardable pair, where n1, n2 touch safe obstacles O1, O2, respectively. Without loss of generality, n1 is a shortest (u, v)-path in the region, and n2 is either another shortest path, or a "second shortest path" as found in Lemma 3.6. (A type 1 move can lead to the first case. A type 2 move can lead to the second case, as we are about to see.) Lemma 4.5. Suppose that the evader is trapped in a type 2 region. Then the third pursuer can guard a path that transitions an obstacle state. Proof. Use Corollary 3.7 to find a guardable (u, v)-path n3 in R[n1, n2] whose homotopy type is distinct from that of both n1, n2. Pursuer p3 establishes a guarding position on n3. The evader is now trapped in either Rp1, n3] or Rp3, n2], so one of O1, O2 transitions from safe to removed. Furthermore, n3 must touch at least one obstacle in each of R[n1, n3] or R[n3, n2]. Otherwise, n3 would be shorter than one of n1, n2, which contradicts the minimality of that path in R[n1, n2]. Depending on the configuration of the obstacles, we may be able to restrict the evader territory further. After doing so, the evader territory may be of any possible type, as shown in Figure 6 (a). □ This brings our discussion to a type 3 region, with p1 guarding a shortest (u, v)-path n1 and p2 guarding a shortest (w, x)-path n2. Our primary goal is to trap the evader in a type 1 region, but we might end up transitioning an obstacle instead. In the latter case, the new evader territory can be of any type, as explained below. Lemma 4.6. Suppose that the evader is trapped in a type 3 region. Then the third pursuer can guard a path so that either (a) the evader is trapped in a type 1 region, or (b) an obstacle transitions to a lower threat level. Proof. Let n3 be a (u, w)-minimal path. Pursuerp3 moves to guard this path using Lemma 3.4. This traps the evader in a smaller region: without loss of generality, this region is bounded by n1, A1, n3. If n3 does not touch any obstacles in this region, then the evader is now in a type 1 region. If n3 touches an obstacle O, then this obstacle transitions to 204 Ars Math. Contemp. 13 (2017) 107-123 Figure 6: Examples of moves where the new guarded path n3 divides the region into five subregions, each identified by its type. (a) A type 2 move. (b) A type 3 move. either safe or removed. The evader could be trapped in a region of any type, as shown in Figure 6 (b). □ We can now prove our main theorem: three pursuers can capture the evader in a two-dimensional compact domain with piecewise analytic boundary. Proof of Theorem 1.2. The first round traps the evader in a type 1' region, or transitions an obstacle state. If we are in a region of type 1, 1' or 2 then we transition an obstacle state in the current round by Lemma 4.4 and Lemma 4.5. When we are in a type 3 region, Lemma 4.6 ensures that we either trap the evader in a type 1 region, or we transition an obstacle. With each path that we guard, the boundary of the updated evader territory is still piecewise analytic by Lemma 4.2. At the end of the round, we update the evader territory and our value for minimum obstacle separation since our new guarded path might be closer to an obstacle than the current value dmin. (Note that the maximum boundary curvature Kmax never increases since all additions to the boundary are line segments.) After at most 2k rounds, we have transitioned all k obstacles to either safe or removed. Once all obstacles have been transitioned, the evader is trapped in a simply connected type 0 region. Lemma 3.5 shows that the evader will then be caught. Each round completes in finite time, so the three pursuers win the game. The capture time upper bound of O(2k • diam(D) + diam(D)2) follows easily. The time required to guard any shortest path is diam(D) by Lemma 3.4 and lion's strategy completes in time diam(D)2 by Theorem 1.1. □ 5 Conclusion In this paper, we described a winning pursuer strategy for a single pursuer in a cat(0) space for turn-based pursuit with capture criterion d(p, e) = 0. We then restricted our attention to compact domains in E2 with piecewise analytic boundary. We showed that three pursuers are sufficient to catch an evader in such environments. By adding a fourth pursuer for use in the final endgame, our strategy could be quickly adapted to a winning strategy in the continuous time version. However, a clever use of the two guarding pursuers during the endgame shows that three pursuers are actually sufficient: see [22] for details. There are plenty of avenues for reseach in geometric pursuit-evasion. Pursuit-evasion results on polyhedral surfaces are an active area of current research [15, 18, 19]. For example, Klein and Suri [15] have proven that 4g + 4 pursuers have a winning strategy on a A. Beveridge and Y. Cai: Pursuit-evasion in a two-dimensional domain 205 polyhedral surface of genus g. Meanwhile, Schroder [20] has proven that at most |_3g/2j + 2 pursuers are needed for a graph of genus g (meaning that such a graph can be drawn on a surface of genus g without edge crossings). It would be natural to consider this question for topological surfaces, or to start by trying to improve the bound for polyhedral surfaces. Likewise, there are a wealth of motion and sensory constraints to consider. Most of these variations of pursuit-evasion have a natural analog in a more general geometric setting. Acknowledgments. This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation. We thank the anonymous referee for helping us to improve the exposition. References [1] M. Aigner and M. Fromme, A game of cops and robbers, Discrete Appl. Math. 8 (1983), 1-12. [2] F. Albrecht and I. D. Berg, Geodesics in euclidean space with analytic obstacles, Proceedings of the American Mathematical Society 113 (1991), 201-207. [3] S. Alexander, R. Bishop and R. Ghrist, Pursuit and evasion in non-convex domains of arbitrary dimensions, in: Proceedings of Robotics: Science and Systems, 2006 . [4] S. Alexander, R. Bishop and R. Ghrist, Capture Pursuit Games on Unbounded Domains, L'Enseignment Mathematique 55 (2009), 103-125. [5] S. Alexander, R. Bishop and R. Ghrist, Total Curvature and Simple Pursuit on Domains of Curvature Bounded Above, Geometriae Dedicata 149 (2010), 275-290. [6] B. Ames, A. Beveridge, R. Carlson, C. Djang, V. Isler, S. Ragain and M. Savage, A leapfrog strategy for pursuit-evasion in polygonal environments, International Journal of Computational Geometry and Applications (to appear). [7] M. Armstrong, Basic Topology, Springer, 1983. [8] D. Bhadauria, K. Klein, V. Isler and S. Suri, Capturing an evader in polygonal environments with obstacles: The full visibility case, International Journal of Robotics Research 31 (2012), 1176-1189. [9] A. Bonato and R. Nowakowski, The Game of Cops and Robbers on Graphs, American Mathematical Society, 2011. [10] M. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Springer-Verlag, 1999. [11] T. H. Chung, G. A. Hollinger and V. Isler, Search and pursuit-evasion in mobile robotics: a survey, Autonomous Robots 31 (2011), 299-316. [12] M. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976. [13] A. Hatcher, Algebraic Topology, Cambridge Univeristy Press, 2002. [14] V. Isler, S. Kannan and S. Khanna, Randomized pursuit-evasion in a polygonal environment, IEEE Transactions on Robotics 5 (2005), 864-875. [15] K. Klein and S. Suri, Pursuit-evasion on polyhedral surfaces, Algorithmica 73 (2015), 730-747. [16] S. Kopparty and C. V. Ravishankar, A framework for pursuit evasion games in Rn, Inf. Process. Lett. 96 (2005), 114-122. [17] J. E. Littlewood, Littlewood's Miscellany, Cambridge Univeristy Press, 1986. [18] N. Noori and V. Isler, Lion and man game on convex terrains, in: Workshop on the Algorithmic Foundations of Robotics (WAFR), 2014 . [19] N. Noori and V. Isler, Lion and man game on polyhedral surfaces with boundary, in: IEEE Conference on Intelligent Robots and Systems (IROS), 2014 . 206 Ars Math. Contemp. 13 (2017) 107-123 [20] B. S. Schroder, The copnumber of a graph is bounded by §genus(G)J + 3, in: J. Koslowski and A. Melton (eds.), Categorical Perspectives: Proceedings of the Conference in Honor of George Strecker's 60th Bitrthday, 2001 pp. 243-263. [21] J. Sgall, A solution to David Gale's lion and man problem, Theoretical Comp. Sci. 259 (2001), 663-670. [22] Z. Zhou, J. R. Shewchuk, H. Huang and C. T. Tomlin, Smarter lions: Efficient full-knowledge pursuit in general arenas, unpublished technical report, University of California, Berkeley, 2012, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1. 381.2253. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 207-225 Spectrum, distance spectrum, and Wiener index of wreath products of complete graphs We describe the adjacency matrix and the distance matrix of the wreath product of two complete graphs, and we give an explicit computation of their spectra. As an application, we deduce the spectrum of the transition matrix of the Lamplighter random walk over a complete base graph, with a complete color graph. Finally, an explicit computation of the Wiener index is given. Keywords: Wreath product of complete graphs, adjacency matrix, distance matrix, spectrum, distance spectrum, Wiener index. Math. Subj. Class.: 05C12, 05C50, 05C76, 05C81, 15A69 1 Introduction The construction of new graphs starting from smaller factor graphs is a very natural and fruitful technique, largely developed in literature for its theoretical interest in several branches of Mathematics - Algebra, Combinatorics, Probability, Harmonic Analysis - but also for its practical applications. Among the standard products we find, for instance, the Cartesian product, the direct product, the strong product, the lexicographic product [22, 23, 30, 31]. More recently, the zig-zag product was introduced [29], in order to produce expanders of constant degree and arbitrary size; in [10, 14], some combinatorial and topological properties of such products, as well as connections with random walks, have been investigated. It is worth mentioning that many of these constructions play an important role in Geometric Group Theory, since it turns out that, when applied to Cayley graphs of two finite groups, they provide the Cayley graph of an appropriate product of these groups (see [1], * Telephone: +39 06 45678356, Fax: +39 06 45678379 E-mail address: alfredo.donno@unicusano.it (Alfredo Donno) Alfredo Donno * Universitá degli Studi Niccold Cusano via Don Carlo Gnocchi 3, 00166 Roma, Italia * Received 31 May 2016, accepted 16 December 2016, published online 3 March 2017 Abstract ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 208 Ars Math. Contemp. 13 (2017) 107-123 where this correspondence is shown for zig-zag products, or [15], for the case of wreath and generalized wreath products). Spectral properties of graph products have been object of an intensive study in the last decades, both for their algebraic and combinatorial interest, and for applications to Probability, Computer Science, and Mathematical Chemistry. The spectrum of a graph is defined as the spectrum of its adjacency matrix; similarly, the distance spectrum of a graph is defined to be the spectrum of its distance matrix (see Section 2). The reader can refer, for instance, to the monograph [5] for an exhaustive treatment of spectra of graphs. We also want to mention the papers [24, 25, 34], where the distance spectrum of some graph compositions has been studied. A related topic of research is the Wiener index, which is defined as the sum of the distances between all the unordered pairs of vertices of the graph. This index was introduced by Wiener [36] and, due to the wide range of applications, it is nowadays largely studied. In particular, it is one of the most frequently used topological indices in mathematical chemistry, as molecules can be represented by means of undirected graphs. For this reason, it has a strong correlation with many physical and chemical properties of molecular compounds, whose properties do not only depend on their chemical formula, but also on their molecular structure [13]. There exists a wide range of fields such as communication, facility location, cryptology, architecture where the Wiener index of a graph is of great interest. A large number of papers is devoted to the study of the Wiener index of graphs, sequences of graphs, products of graphs. In [12] the Wiener index of trees is investigated. In [16] the Wiener index and the related Hosoya polynomial are studied for a family of circulant graphs. See also the paper [9], where the Wiener index is studied on an increasing sequence of finite graphs, introduced in [6], and whose limit graphs have been studied in [7], which approximates the Sierpinski carpet fractal. In [17, 18] the study of Wiener index is developed for some graph compositions. In the present paper, we focus our attention on a different kind of graph product known in literature, namely the wreath product of two graphs (see Definition 2.1). This construction is nowadays largely studied, and different generalizations have been introduced [15, 19]. Notice that this construction is interesting not only from an algebraic and combinatorial point of view, but also for its connection with Geometric group theory and Probability, via the notions of Lamplighter group and Lamplighter random walk (see, for instance, [3, 21, 32, 33, 37]). Notice that in a previous paper joint with D. D'Angeli [8], we introduced a matrix operation, called wreath product of matrices (recalled in Definition 2.2), which is a matrix analogue of the wreath product of graphs, since it provides the adjacency matrix of the wreath product of two graphs, when applied to the adjacency matrices of the factors (Theorem 2.3 below). Let us denote by Kn the complete graph on n vertices. In this paper, the wreath product Kn l Km is studied. In Proposition 3.1, we describe in detail distances in Kn l Km, and we deduce its diameter in Corollary 3.2. Moreover, in Proposition 3.4 we show that the graph Kn l Km is not distance-regular. After describing in detail the adjacency matrix of the wreath product Kn l Km of two complete graphs, we are able to explicitly compute its spectrum by using a reduction argument, allowing to reduce our computations to the study of the spectrum of smaller matrices, whose size is the cardinality of the vertex set of the first graph (Theorem 3.7); we then deduce the spectrum of the transition matrix of the Lamplighter random walk with base graph Kn and color graph Km (Corollary 3.9). In Proposition 3.10, we provide the distance matrix of KnlKm, and its spectrum is determined A. Donno: Spectrum, distance spectrum, and Wiener index of wreath products of. 209 in Theorem 3.11 by means of a second reduction argument. Finally, in Theorem 3.13, the Wiener index of the graph Kn I Km is computed. Notice that the spectrum considered in the present paper concerns the "walk or switch" Lamplighter random walk. The analogous question for the so called "switch-walk-switch" Lamplighter random walk has been solved in [26, 27]. A common framework for such computations has been established in [20]. 2 Preliminaries Let G = (V, E) be a finite undirected graph, where V denotes the vertex set, and E is the edge set consisting of unordered pairs of type {u, v}, with u, v e V. If {u, v} e E, we say that the vertices u and v are adjacent in G, and we use the notation u ~ v. A path in G is a sequence u0, ui,..., u of vertices such that u ~ ui+1, for each i = 0,..., I - 1. We say that such a path has length I. The graph is connected if, for every u, v e V, there exists a path u0, u1,..., u^ in G such that u0 = u and u = v. For a connected graph G, we will denote by d(u, v) the geodesic distance between the vertices u and v, that is, the length of a minimal path in G joining u and v. The diameter of G is then defined as diam(G) = max„,„Ev {d(u, v)}. Recall now that the adjacency matrix of an undirected graph G = (V, E) is the square matrix A = (au,v )u,vev, indexed by the vertices of G, whose entry au,v equals the number of edges connecting u and v. As the graph G is undirected, A is a symmetric matrix and so all its eigenvalues are real. The spectrum of G is then defined as the spectrum of its adjacency matrix. The degree of a vertex u e V is defined as deg(u) = J2V£V au,v. In particular, we say that G is regular of degree d, or d-regular, if deg(u) = d, for each u e V. In this case, the normalized adjacency matrix A' of G is obtained as A' = d A. We recall now the definition of wreath product of graphs. Definition 2.1. Let G1 = (V1, E1) and G2 = (V2, E2) be two finite graphs. The wreath product G11 G2 is the graph with vertex set V2Vl x V1 = {(f, v) | f: V1 ^ V2, v e V1}, where two vertices (f, v) and (f', v') are connected by an edge if: (1) (edges of type I) either v = v' =: v and f (w) = f '(w) for every w = v, and f (v) ~ f'(v) in G2; (2) (edges of type II) or f (w) = f' (w), for every w e V1, and v ~ v' in G1. It follows from the definition that, if G1 is a regular graph on n1 vertices with degree d1 and G2 is regular graph on n2 vertices with degree d2, then the graph G11G2 is a (d1 + d2 )-regular graph on n^1 vertices. It is a classical fact (see, for instance, [37]) that the simple random walk on the graph G11G2 is the so called Lamplighter random walk, according to the following interpretation: suppose that at each vertex of G1 (the base graph) there is a lamp, whose possible states (or colors) are represented by the vertices of G2 (the color graph), so that the vertex (f, v) of G1 I G2 represents the configuration of the |V1| lamps at each vertex of G1 (for each vertex u e V1, the lamp at u is in the state f (u) e V2), together with the position v of a lamplighter walking on the graph G1 . At each step, the lamplighter may either go to a neighbor of the current vertex v and leave all lamps unchanged (this situation corresponds to edges of type II in G11 G2), or he may stay at the vertex v e G1, but he changes the state of the lamp which is in v to a neighbor state in G2 (this situation corresponds to edges of type I in G11 G2). For this reason, the wreath product G11 G2 is also called the Lamplighter 210 Ars Math. Contemp. 13 (2017) 107-123 graph, or Lamplighter product, with base graph Gi and color graph G2. Also notice that the model described above is often called "walk or switch" Lamplighter random walk. It is worth mentioning that the wreath product of graphs represents a graph analogue of the classical wreath product of groups [28], as it turns out that the wreath product of the Cayley graphs of two finite groups is the Cayley graph of the wreath product of the groups, with a suitable choice of the generating sets. In the paper [15], this correspondence is proven in the more general context of generalized wreath products of graphs, inspired by the construction introduced in [2] for permutation groups. Also notice that in [19] a different notion of generalized wreath product of graphs is presented. In the paper [8], the following matrix construction involving wreath products is introduced. Let Mmxn(C) denote the set of matrices with m rows and n columns over the complex field, and let in be the identity matrix of size n. We recall that the Kronecker product of two matrices A = (a^)i=i,...,m;j=i,...,„ € Mmx„(C) and B = (bhk)h=i,...,p-,k=i,...,q € Mpxq(C) is defined to be the mp x nq matrix ( aiiB • • • ainB A ® B = . ... . \ amiB • • • amnB We denote by A0" the iterated Kronecker product A < • • • < A, and we put A0° = 1. Definition 2.2 ([8]). Let A € Mnxn(C) and B € Mmxm(C). For each i = 1,..., n, let Ci = (chk)h,k=i,...,n € Mnxn(C) be the matrix defined by = 1 if h = k = i chk = ^ 0 otherwise. The wreath product of A and B is the square matrix of size nmn defined as n A i B = 10" < A + £ if-1 << B << << Ci. i=i In [8] the following theorem, which shows the correspondence between wreath products of matrices and wreath products of graphs, is proven. Theorem 2.3. Let A[ be the normalized adjacency matrix of a di-regular graph Gi = (Vi,Ei) and let A2 be the normalized adjacency matrix of a d2-regular graph G2 = (V2,E2). Then the wreath product A'^ i A2^ is the normalized adjacency matrix of the graph wreath product G' iG2. For a finite connected graph G = (V, E), the distance matrix D = (du,v)u,vev of G is, by definition, the square matrix indexed by the vertices of G, such that du,v = d(u, v). The matrix D is symmetric by definition, so that its spectrum is real. The spectrum of D is usually called the distance spectrum of the graph G. We complete this preliminary section by recalling the definition of Wiener index of a finite connected graph G = (V, E). The Wiener index of G is defined as the sum of the distances between all the unordered pairs of vertices, i.e., W(G) = 1 £ d(u,v), u,vG V n times A. Donno: Spectrum, distance spectrum, and Wiener index of wreath products of. 211 where d(u, v) denotes the usual geodesic distance between u and v. In Section 3, we will construct the adjacency matrix and the distance matrix of the graph Kn I Km, and we will compute their spectra. Finally, we will provide an explicit computation of the Wiener index of Kn I Km. 3 The wreath product Kn I Km From now on, we will focus our attention on the wreath product Kn I Km, where Kn (Vn, En) is the complete graph on n vertices, that is, the graph on n vertices in which every pair of distinct vertices is connected by a unique edge. Notice that Kn is a regular graph of degree n - 1, with diameter 1, where d(u, v) = j ^ if U = V for each pair u, v of vertices. Figure 1: The complete graph K6. In particular, the adjacency matrix of Kn is given by Adn = Jn - /n, where Jn denotes the uniform square matrix of size n, whose entries are all equal to 1. Moreover, it follows from Theorem 2.3 that the adjacency matrix of the graph Kn I Km is the matrix n Adn I Adm = /®" ® Adn + ^ if" ® Adm /®"-< ® Cj, (3.1) j=1 with C as in Definition 2.2. Notice also that, by definition, Kn I Km is an (n + m -2)-regular graph on nmn vertices. A vertex of Kn I Km will be usually denoted by (y1,..., yn)xj, where yj G Vm, for each j = 1,..., n, and xj G Vn. In the lamplighter interpretation, we can think that the lamp placed at the j-th vertex xj of Kn has color yj, with yj G Vm, and the lamplighter is in position xj. Moreover, it follows from the definition of wreath product of graphs that two vertices u = (y1,..., yn)xj and v = (y'x,..., y'n)xk have distance 1 if either there exists a unique index j G {1,..., n} such that yj = yj and xj = xk; or yj = yj for each j, and xj = xk (observe that xj ~ xk in Kn if and only if xj = xk, as the graph Kn is complete). We are going to develop an explicit analysis of the variability of the distances between two vertices in the graph Kn I Km. Let u = (y1,..., yn)xj and v = (y',..., y'n)xk be two vertices of Kn ^ Km. Put J = {1, 2,..., n} and define the partition J = U JUjV by = {j G J : yj = yj } Jl,„ = {j G J : yj = yj }. (3.2) Note that the cardinality | J^ v | can be interpreted as the Hamming distance between the "lamp strings" (y1,..., yn) and (y',..., yn). The following proposition holds. 212 Ars Math. Contemp. 13 (2017) 107-123 Proposition 3.1. Let u = (yu,..., yn)xi and v = (y[,..., y'n )xk be two vertices of Kn I Km and let J0 v and J^ v as in (3.2). Then d(u, v) 0 if i = k . i 1 if i = k J Ju 'v 1 if i = k = j* 0; d(u,v) = < 3 if i = j* = k 2 if i = j* = k; or i = j* = k More generally, for 2 < \J1v | < n: if Ju,v = {j*}. d(u,v)=\ 2\Ju 2\JU,v\ + 1 if k,i e J0,v if i e J0,v ,k e JU,v ; or i e JU,v ,k e JU 2\JU.v\- 1 + Sik if i,k e Ju J1 u,v 0 u,v with Sik = 1 if i = k 0 if i = k. Proof. First of all observe that, if = 0, we have yj = yj for each j e J, so that u and v coincide if i = k, whereas they are adjacent, by an edge of type II in Kn I Km, if i = k. Suppose now J^v = {j*}. In the first case, the vertices u = (yu,... ,yjt,..., yn)xjt and v = (yi,...,yj t,..., yû)xjt, with yjt = y'jt, are adjacent in Kn I Km. In the second case, when i = j* = k, the path (yi,...,yjt,...,yn)xi ~ (yi,...,yjt,...,yn)xjt ~ (yi,...,yj„,...,yn) xj - ivi, ...,v'j,,.. .,Vn)xh is a path of minimal length joining u and v. In the third case, when i = j* = k, the path (Vl,...,Vj, ,...,Vn)xj, - (vi,...,vj, ,...,Vn)xj, - (vi,...,v'j, , ... , Vn)xk is a path of minimal length joining u and v; the case i = j* = k is similar. Now let 2 < | J | = h < n, with v = {j1,... ,jh}. In other words, the n-tuples (y1,... ,yn) and (y[,... ,y'n) differ exactly in h places, indexed by the elements j1,... ,jh of . In the first case, the path (Vi,...,Vji ,...,Vn)xi - (Vi,...,Vj1 ,...,Vn)xji - (Vi,...,y'j1 ,...,Vn )xji - - (vi,...,v'j 1,..., Vn)xj2-----(vi,...,v'j 1,..., vj h_i ,..., Vn)xjh - - (y1,...,y'jl ,...,v'jh_i ,...,v'jh , ... , yn)xjh -- (y1,...,y'j i ,...,yj h_ i ,...,yjh ,...,yn)xk is a minimal path joining u and v, and it has length 2h +1. In the second case, when i e J0v, k e J^ v, we can assume, without loss of generality, because Kn is complete, that jh = k, so that the last step is not necessary, and a path of minimal length connecting u and v has length 2h; a similar argument works in the case i e J^ v, k e J0 v. Finally, if i = k and i,k e J^v, we can assume that j1 = i and jh = k. Now, a path of minimal A. Donno: Spectrum, distance spectrum, and Wiener index of wreath products of. 213 length joining u and v is given by (yi,...,yji ,...,Vn)xj1 - (yi,...,yj i ,...,y„)xj1 - (yi,...,yj i,..., y„ )a - (yi,..., yj1,..., yj2, • • •, yn)xj2 ----- (yi, •• •, yji,...,yj2 •.-,yjh_i, • • • ,yn)xjh-i - - (yi,...,yji,---,yj2 -•-,yjh_i, ...,yn)xjh -- (yi,---,yji, • • • ,yj h-i ,---,yjh, • • • , yn)xjh and has length 2h - 1. In the special case i = k, we need one more step in order to reach the final vertex xk = xj, by means of an edge of type II in Kn I Km. □ Corollary 3.2. The diameter of the graph Kn ? Km is 2n. Proof. It follows from the proof of Proposition 3.1 that the maximal distance d(u, v) between two vertices u and v of Kn ^ Km is equal to 2n, and it is obtained when the vertices u, v have the form u = (yi, • • •, yn)xj v = (yi, .. ., yn)xfc, with yj = yj, for each j = 1,..., n and xj = xk. In fact, we get in this case d(u, v) = 2| Ji,„ | - 1 + Sjk = 2n - 1 + 1 = 2n. □ Now, for each i = 0,1,..., 2n, and every vertex u of Kn I Km, we denote by Sj(u) the sphere of radius i centered at u, that is: Sj(u) = {v € V(Kn I Km) : d(u,v) = i}. Because of the complete symmetry of the graph, it is clear that the integer s j = |Si(u)| does not depend on the particular choice of the vertex u. We recall below the classical definition of distance-regular graph (see, for instance, [4], or [5, Chapter 12] for some results about spectral properties of distance-regular graphs, also in connection with association scheme theory). Definition 3.3. A connected graph G is said to be distance-regular if it is regular and, for any two vertices u, v at distance i, there are exactly cj neighbors of v in Sj-i(u) and bj neighbors of v in Sj+i(u). If d is the diameter of G, the sequence {b0, bi,..., bd-i; ci, c2,..., cd} is usually called the intersection array of G; notice that the integers c0 and bd are undefined. Proposition 3.4. For every n, m, the wreath product Kn I Km is not distance-regular. Proof. Consider two vertices of type u = (yi,..., yn)xj and v = (yi,..., yn)xj, with j = i, so that d(u, v) = 1. Now, the neighbors of v having distance 2 from u are exactly the vertices of type (yi,..., yj,..., yn)xj, with yj = yj: the number of such vertices is m - 1. On the other hand, consider the vertex w = (yi,..., yj,..., yn)xj, with yj = yj, so that we still have d(u, w) = 1. It is clear that the neighbors of w having distance 2 214 Ars Math. Contemp. 13 (2017) 107-123 from u are exactly the vertices of type (y^ ..., yj,..., yn)xj, with xj = xj, and they are precisely n - 1. This implies that the coefficient b\ cannot be defined, and this is sufficient to conclude that Kn i Km is not distance-regular. Also in the case n = m, the graph is not distance-regular. In order to show that, it suffices to consider the vertices u = (yi,..., yn-i, yn)xn and v = (yi,..., y'n_1,yn)xn, with yj = yj for each j = 1,..., n - 1, so that d(u, v) = 2n - 1, according to Proposition 3.1. Now, the neighbors of v having distance 2n from u are exactly the vertices of type (y',..., yn-i, y'n)xn, with y'n = yn: the number of such vertices is n - 1. On the other hand, consider the vertices w = (yi,..., yn)xj and z = (y',..., y'n)xj, with yj = yj for each j = 1,..., n and xj = xj, so that we still have d(w, z) = 2n - 1. In this case, the unique neighbor of z having distance 2n from w is the vertex (y',..., y'n)x^ This implies that the coefficient b2n-i cannot be defined, and this is sufficient to conclude that Kn i Kn is not distance-regular. □ Example 3.5. Consider the graph K3 i K2 depicted in Figure 2, where the vertices of K3 and K2 are identified with the sets {0,1,2} and {0,1}, respectively. The adjacency matrices of the graphs K3 and K2 are, respectively, 011 Ad3 = ( 1 0 1 1 1 0 and Ad2 = 01 10 so that the matrix wreath product Ad3 I Ad2 = if3 Ad3 + Ad2 /2 /2 C + + /2 Ad2 ® /2 ® C2 + /2 ® /2 ® Ad2 ® C3 is the adjacency matrix of the graph K3IK2. The graph K3 ^ K2 is regular of degree 3, and its diameter is 6. 3.1 Spectrum of the graph Kn I Km In this section, we will give an explicit description of the spectrum of the graph Kn I Km which is, by definition, the spectrum of its adjacency matrix Adn I Adm described in Equation (3.1). In order to develop our analysis, we need to recall the definition of circulant matrix. A (complex) circulant matrix C of size m is a square matrix with m rows and m columns, of type C i co ci Cm-1 Co Ci c1 cm1 cm -1^ c1 co with Ci e C, Vi = 0,..., m - 1. (3.3) The reader can refer to [11] as an exhaustive monograph on circulant matrices. The following theorem has been proven in [8], by using the spectral analysis developed in [35] for block circulant matrices. A. Donno: Spectrum, distance spectrum, and Wiener index of wreath products of. 215 000, 0 100, 0 000,1 010, 1 .000, 2 001,1 011, 1 011, 2 010, 0 100, 2 001, 0 101, 0 001, 2 >--P-«r-f 101, 2 101, 1 111, 1 111, 2 011, 0 111, 0 010, 2 110, 2 110, 0 100, 1 110, 1 Figure 2: The graph K3 I K2. Theorem 3.6. Let A be a square matrix of size n, and let B be a circulant matrix of size m as in (3.3). Then the spectrum E of the matrix A I B is obtained by taking the union of the spectra of the mn matrices of size n given by M n'i2' A + EE ciPUtCt, t=1 i=0 where ij G {0,1,... ,m — I}, for every j = 1,... ,n, and p = exp (. In particular, Theorem 3.6 can be applied in order to determine the spectrum of the adjacency matrix Adn I Adm = /m" ® Adn + ^ Z^ ' ® Adm ® ® Ci, i=i since the matrix Adm is a circulant matrix, with c0 = 0 and ci = 1, for each i = 1,... ,m-1. When listing eigenvalues and their multiplicities in the next theorem, and in the rest of the paper, we will write Ah to say that the eigenvalue A has multiplicity h; the multiplicity will be omitted when it is equal to 1. We obtain the following result. n (n)(m- 1)" Theorem 3.7. The spectrum E of the graph Kn I Km is E = |Jn=0 Ek n-1. with Eo = {(-2)n-1; n - 2} Efc = (m - 2)k ; (-2) k— 1 ( o^n— k — 1 m+n-(m-n)2 +4 km k = 1, .. . ,n — 1 2 216 Ars Math. Contemp. 13 (2017) 107-123 and £„ = {(m - 2)n-1; m + n - 2} . Proof. By virtue of Theorem 3.6, the spectrum of Kn I Km is obtained by taking the union of the spectra of the matrices n m— 1 M il,i2,...,i n _ Adn + E £ CipiitCt, t=1 i=0 where ij e {0,1,..., m - 1}, for each j = 1,..., n, and p = exp (. Notice that c0 = 0 and ci = 1 for each i = 1,...,m — 1. Moreover, the following identity holds: m— 1 E i=1 m1 (pi4) = \ (p* )m—1 pi t—1 if it =0 - 1 = -1 if it =0 since p is an m-th root of unity. Therefore, the matrix Mii,i2>...>in can be rewritten as M11 ,12,...,1 n Adn + E (m - 1)Ct - E Ct t:i t=0 t:it = 0 = Jn - In + E (m - 1)Ct - ( E Ct + E Ct ) + E Ct t:it=0 t:it=0 t:it=0 t:it=0 = Jn - 21 n + m E Ct. t:it=0 By using iterated conjugations with appropriate elementary permutation matrices, it can be shown that the spectrum of the matrix M11'12' . '1" only depends on the number k of indices equal to 0 in the n-tuple (i1, i2,..., in), but it is independent of the particular position of such indices. As a consequence, for each k = 0,1,..., n, we can reduce to investigate the spectrum of the matrix M0'...'0'ik+i'...,in, corresponding to the n-tuple (0,..., 0, ik+1,..., in), with ij = 0 for each j = k + 1,..., n. We have: k times ( -1 + m 1 M 0'...'0'1k + 1'...'1n 1 Then we can write M^-A«^1'...'1" 1 -1 + m 1 1 1 1 -1 Jn + Q, where Q = m^t=1 Ct - 2/n is the 1 1 1 1 1 A. Donno: Spectrum, distance spectrum, and Wiener index of wreath products of. 217 diagonal matrix ( -2 + i Q = -2 + m 2 -2 Now we have: det(A/n - MO-.A^+i-.^) = det (A/„ - J„ - Q) det ((A/„ - Q) (/„ - (A/„ - Q)-1 J„)) det(A/„ - Q) • det (/„ - (A/„ - Q)-1 J„) It is clear that det(A/„ - Q) = (A - (m - 2))k • (A + 2) n — k (3.4) Now it can be seen that the matrix (A/n - Q) 1 Jn is the matrix of rank 1, whose first k rows are constant, with entries all equal to A—^_2), whereas the remaining n - k rows are constant, with entries all equal to a+2. Therefore, (A/„ - Q)_1 Jn has n - 1 eigenvalues equal to 0, and one eigenvalue equal to x_mi_2) + x+f. This implies that the matrix In - (A/„ - Q)_1 Jn has n - 1 eigenvalues equal to 1, and one eigenvalue equal to 1 -—T—or — T_f, so that: A —(m-2) A+2 ' det (In - (A/n - Q) — 1 Jn) = 1 - n — k A - (m - 2) A + 2' (3.5) By gluing together (3.4) and (3.5), we obtain: det(A/n - MOv-Ai^1,..,^) = (a - (m - 2))k-1 • (A + 2) n- k- 1 (A2 + (4 - m - n)A + mn + 4 - km - 2n - 2m). For the particular value k = 0, we get: det(A/n - M) = (A + 2)n-1 • (A - (n - 2)); for the particular value k = n, we have: det(A/n - M0,...,0) = (A - (m - 2))n-1 • (A - (m + n - 2)). The claim follows, if we observe that, for each k = 0,1,..., n, the spectrum of £k must be considered (T) • (m - 1)n—k times, corresponding to the number of n-tuples (i1,..., in) with k indices equal to 0, and the remaining indices varying in {1,...,m - 1}. □ k 218 Ars Math. Contemp. 13 (2017) 107-123 Example 3.8. Consider the graph K3 l K4, so that n — 3 and m — 4. The spectrum of the matrix Ad3 l Ad4 consists of the following eigenvalues: 5; 211; 127; (-2)81; (^V. The corresponding matrices M11 ,i2'®3 of size 3, with i1, i2, i3 G {0,1, 2, 3}, have eigenvalues: (a) (-2)2; 1, for k — 0. For instance, this is the case of the matrix -1 1 1 M1-1-1 — J3 - 2/3 — | 1 -1 1 1 1 -1 (b) -2; , for k — 1. For instance, this is the case of the matrix 3 1 1 M0-1'1 — J3 - 2/3 + 4C1 — | 1 -1 1 1 1 -1 (c) 2; 3±233, for k — 2. For instance, this is the case of the matrix 3 1 1 M°'°'1 — J3 - 2/3 + 4(C1 + C2) — ( 13 1 V1 1 -1 (d) 22; 5, for k — 3. This is the case of the matrix __ 3 11 M— J3 - 2/3 + 4(C1 + C2 + C3) — J3 + 2/3 — ( 1 3 1 113 Corollary 3.9. The spectrum E' of the transition matrix of the Lamplighter random walk with base graph Kn and color graph Km is e' — u n=° £'fc(k)(m-1r- , with \ n-1 E' — ) I 2 \ . n—2 ° | \ m+n —2 J ; m+n—2 E' — / ( m — 2 \k—1 . (_ 2 \n—k—1 , m+n—4^(m—n)2+4fcm Ek 1 ^m+n — 2 J ; ^ m+n—2 J ; 2(m+n—2) for k — 1,..., n - 1, and \ n— 1 — < ; 1 Proof. It suffices to take into account that the transition matrix of the Lamplighter random walk on the base graph Kn, with color graph Km, is the normalized adjacency matrix of the graph Kn l Km, which is a regular graph of degree m + n - 2. □ A. Donno: Spectrum, distance spectrum, and Wiener index of wreath products of. 219 3.2 Distance spectrum and Wiener index of the graph Kn I Km The aim of this section is to describe the distance matrix of the graph Kn l Km, together with its spectrum. Moreover, we will exhibit an explicit computation of the Wiener index of the graph. Proposition 3.10. The distance matrix of the graph Kn i Km is the matrix D = (i E ,i„)e{0,i}n Adm Adm ■ ■ ■ Adm A. il,i2, (3.6) where we put AdPm = Im, and the matrix Ai1,i2,...,in is the square matrix of size n, indexed by the vertices of Kn, defined as follows. Let {ii,..., in} = I0 U Ii, with I0 = {ij : ij = 0} and Ii = {ij : ij = 1}. Then, for any pair of vertices xi and xk of Kn: (a) Ah,...,in = Adn = Jn - In if Ii = 0/ 1 if i = k = j 3 if i = j* = k 2 if i = j* = k; or i = j* 2|Ii| + 1 if i,k e Io 2|Ii| if i e I0,k e Ii; or i e Ii,k e I0 21Ii | — 1 + Sik if i,k e Ii 1 if i = k 0 if i = k. (b) Ai1,...,in(xi,xk) (c) Ai1,...,in(xi,xk) if h = {ij,}; k if 2 < |I1| < n, where Sik Proof. Observe that, for each j = 1,..., n, the index ij G {0,1} establishes whether the color of the lamp at the j-th vertex xj of Kn is changed. More precisely, the index ij = 0 produces the matrix AdPm = Im as j-th term of the Kronecker product, so that we are not changing the color of the lamp in that position; conversely, the index ij = 1 provides the matrix Adm as j-th term of the Kronecker product, so that we are changing the color of the lamp in that position, with any other color, as Km is the complete graph. Therefore, for any fixed n-tuple (i1, ...,in ) G {0,1}n, the contribution Adm ®Adm ■ -<8>Adm ®Aiui2..in to D must take into account the distances between vertices u, v of Kn i Km corresponding to lamp configurations which differ exactly at the places indexed by I1 . Therefore, if the configurations of lamps corresponding to the vertices u = (y1,... ,yn)xi and v = (y1,..., y'n)xk of Kn i Km differ at exactly |I11 vertices of Kn, indexed by I1, the last contribution in the Kronecker product is an n x n matrix, whose entry (xi, xk) must be equal to the distance d(u, v). Then the claim follows from Proposition 3.1. □ As in the case of the adjacency matrix Adn i Adm, the spectrum of the matrix D can be computed by using a reduction argument. In fact, the matrix D in (3.6) has the following block circulant structure D0 Dm D D1 Do D m— 1 \ D0 D1 D m1 D1 Do ••• Dm—1 D1 D0 (3.7) 220 Ars Math. Contemp. 13 (2017) 107-123 with Do = £ Adim Adm < Ao,i2,...,i„ (¿2,...,i„)e{o,i}n Dj = ^ Adii Adim < Ai,i2,...,in for each i = 1 ,...,m - 1. (¿2,...,i„)e{o,i}n Then the spectral analysis of block circulant matrices developed in [35] ensures that the spectrum of D can be obtained by taking the union of the spectra of the following m matrices of size nmn-i: m— i Djl = £ phljl Dhl hi=0 = £ Adii «•••<< Adim < Ao,i2,...,i„ + (¿2,...,in)e{0,i}n i —i + £ Phljl £ Adii Adii < Ai,i2,...,i„ hi=i (¿2,...,i„)e{o,i}n (i—i \ Ao,,2,.,j„ + £ PhljlAi,j2,...,jJ , . hl = i J with j € {0,1,..., m - 1}. Observe that each of these matrices is still a block circulant matrix, with blocks of size nmn—2, given by (rn— i \ Ao,o,ia,...,i„ + £ PhljlAi,o,i3,...,iJ , hl=i (rn— i \ Ao,i,i3,...,i„ + £ PhljlAi,i,i3,...,i„ hl=i for i = 1, . . . , m - 1. Therefore, the same argument can be repeated, so that the spectrum of D is obtained by taking the union of the spectra of the following m2 matrices of size n2 nmn 2: i— i Djl,j2 = £ ph2j2 Dh2 = h2=o (i—i \ Ao,o,i3,...,i„ + £ PhljlAi,o,i3,...,iJ + . , . , , hl = i J i— i + £ ph2j2 £ Adii Adii < h2=i (¿3,...,i„)e{o,i}n ii < Ao,i,i3,...,i„ + £ phljlAi,i,i3,...,i„ hl = i A. Donno: Spectrum, distance spectrum, and Wiener index of wreath products of. 221 E Adm ® ■ ■ ■ ® Adm ® i ^0,0,^3 ,...,in+ (¿3,...,i„)e{0,i}n V m—1 m—i + £ /iji Ai,0,i3,...,i„ + E Ph2j2 A0,1,i 3,...,in + hi = 1 h2 = 1 m—1 m—1 ^ + E Phljl E Ph2j2 A1,1,i3,...,in hi = 1 h2 = 1 / with (j1, j2) € {0,..., m - 1}2. This reduction argument can be iterated further, until we get blocks of size n. Once again, notice that ^mT/ Phjs = j m_1 1 if js =0 We thus have proven the following theorem. Theorem 3.11. The distance spectrum E of the graph Kn l Km is obtained by taking the union of the spectra Ej1,...,jn of the mn matrices of size n: n /m— 1 \ is Djl,.j = E n E PhSjS Aii,...,i„ , (ii ,...,i„)e{0,1}n S=1 Vhs = 1 J (j 1,..., jn) €{0,...,m - 1}n, where, if we put I0 = {ij : ij = 0} and I1 = {ij : ij = 1}, we have: A0,...,0 = Adn; {1 if i, k € I1 3 if i,k € I0 for |I1| = 1; 2 if i € I1, k € I0; or i € I0, k € I1 and ( 2|I1| + 1 ifi,k € I0 Aii,...,in (xi,xk) = < 21111 if i € I0,k € I1; i € I1,k € I0 [2|A|- 1 + Sik ifi,k € I1 for 2 < |i1| < nwith Sik = j 0 ifi=k Example 3.12. Let us consider the explicit example K3 l K3. The distance matrix of this graph is D = I3 ® I3 ® I3 ® A000 + I3 ® I3 ® Ad3 ® A001 + I3 ® Ad3 ® I3 ® A010+ + I3 ® Ad3 Ad3 A011 + Ad3 I3 I3 A100 + Ad3 I3 Ad3 A101 + + Ad3 ® Ad3 ® I3 ® A110 + Ad3 ® Ad3 ® Ad3 ® Am, with 0 1 1 3 3 2 A000 = | 10 1 I A001 = | 3 3 2 1 1 0 2 2 1 222 Ars Math. Contemp. 13 (2017) 107-123 A, 010 A 100 A 110 3 2 3 2 1 2 323 122 233 233 4 3 4 3 4 4 4 4 5 A 011 A 101 A 111 The spectrum of D consists of the following eigenvalues: 544 443 434 443 454 344 6 5 5 5 6 5 5 5 6 312; 152; 048; (-3)18; (-24 ± 3V43)6 We have, for instance: D0,2,0 = A000 + 2A001 - A010 - 2A011 + 2A100 + 4A101 - 2A110 - 4Am = -21 -9 -18 -9 -9 -9 -18 -9 -21 whose eigenvalues are -3 and -24 ± 3v/43. Next, we pass to the computation of the Wiener index W(Kn l Km) of the graph Kn l Km. It follows from the definition of the Wiener index that W (Kn l Km) is given by the sum of all the entries of D, divided by 2, due to the fact that each contribution d(u, v) appears twice, as the matrix D is symmetric. Keeping in mind the block structure of the distance matrix D described in (3.7) and the fact that each block Dj, for i = 0,..., m -1, appearing in (3.7) can be recursively regarded as a block circulant matrix, until one gets elementary blocks of size n represented by matrices of type Ailv..,in, we obtain the following result. Theorem 3.13. The Wiener index of the graph Kn l Km is W(Kn l Km) = nmn(2mnn2 - nmn - 2n2mn-1 + mn + 2nmn-1 - mn-1 - m). Proof. First of all, for every n-tuple (i1,...,in) G {0,1}n, put: dj ■i1,...,i„ Ail,...,in (x Xi,Xj G Vn ). Now observe that, by definition of the matrices Ail,...,in, the sum dil,...,in only depends on the cardinality of the sets 10 = {i3 : i3 = 0} and 11 = {i3 : i3 = 1}, while it is independent from the particular position of the indices. Therefore, for every k = 0,1,..., n, it makes sense to define: dfc = Y^ A1, ..., 1,0,..., 0(xi,x3). Xi,Xj G Vn k times n-k times Moreover, by performing a direct computation which uses the explicit description of the matrices Ail,...,in given in Theorem 3.11, we are able to determine: A. Donno: Spectrum, distance spectrum, and Wiener index of wreath products of. 223 (a) d0 = n(n — 1); (b) di = 1 + 4(n — 1) + 3(n — 1)2; (c) dk = (2k + 1)(n — k)2 +4k2(n — k) + 2k2 + k(k — 1)(2k — 1) for 2 < k < n. Now we have to establish the number of contributions of type dk to W(Kn I Km), for every k. First of all, a factor equal to (n) appears, taking into account all the possible choices of k indices equal to 1. Moreover, a second factor given by mn(m — 1)k appears, since a fixed n-tuple (i1,... ,in) containing k indices equal to 1 (see Equation (3.6)) produces mn (m — 1)k blocks of size n, within the matrix D, which are equal to Aj j , due to the fact that, when we change the color of a lamp, we have m — 1 possibilities for the choice of the new color. This implies that Example 3.14. 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ARS MATHEMATICA CONTEMPORANEA 13 (2017) 227-234 Counting faces of graphical zonotopes Vladimir Grujic * University of Belgrade, Faculty of Mathematics, Studentski trg 16, Belgrade, Serbia Received 15 June 2016, accepted 16 January 2017, published online 6 March 2017 Abstract It is a classical fact that the number of vertices of the graphical zonotope Zr is equal to the number of acyclic orientations of a graph r. We show that the f -polynomial of Zr is obtained as the principal specialization of the q-analog of the chromatic symmetric function of r. Keywords: Graphical zonotope, f -vector, graphical matroid, symmetric function. Math. Subj. Class.: 05E05, 52B05, 16T05 1 Introduction The f -polynomial of an n-dimensional polytope P is defined by f (P, q) = ^ "=o fi(P)qi, where fi(P) is the number of ¿-dimensional faces of P. The f-polynomial f (Zr,q) of the graphical zonotope Zr is a combinatorial invariant of a finite, simple graph r. The vertices of Zr are in one-to-one correspondence with regions of the graphical hyperplane arrangement Hr, which are enumerated by acyclic orientations of r. Stanley's chromatic symmetric function ^(r) = J2 f proper Xf of a graph r = (V, E), introduced in [7], is the enumerator function of proper colorings f: V ^ N, where Xf = xf (!) • • • x f (n) and f is proper if there are no monochromatic edges. The chromatic polynomial x(r, d) of the graph r, which counts proper colorings with a finite number of colors, appears as the principal specialization The number of acyclic orientations of r is determined by the value of the chromatic polynomial x(r, d) at d = -1, [6] * Author is supported by Ministry of Education, Science and Technological developments of Republic of Serbia, Project 174034. E-mail address: vgrujic@matf.bg.ac.rs (Vladimir Grujic) x(r,d) = ps(*(r))(d) = *(r) | al (r) = (-i)|V |x(r,-1). (1.1) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 228 Ars Math. Contemp. 13 (2017) 107-123 There is a q-analog of the chromatic symmetric function (T) introduced in a wider context of the combinatorial Hopf algebra of simplicial complexes considered in [2]. It is a symmetric function over the field of rational functions in q. The principal specialization of (r) is the q-analog of the chromatic polynomial xq (r, d). The main result of this paper is the following generalization of formula (1.1): Theorem 1.1. Let r = (V, E) be a simple connected graph and Zr the corresponding graphical zonotope. Then the f -polynomial of Zr is given by f (Zr,q) = (-l)|V|x-q(r, -1). The cancellation-free formula for the antipode in the Hopf algebra of graphs, obtained by Humpert and Martin in [3], reflects the fact that f (Zr, q) depends only on the graphical matroid M(r) associated to r. For instance, for any tree Tn the graphical matroid is the uniform matroid M (Tn) = Un and the corresponding graphical zonotope is the cube ZTn = In-1. Whitney's theorem from 1933 describes how two graphs with the same graphical matroid are related [9]. It can be used to find more interesting nonisomorphic graphs with the same f -polynomials of corresponding graphical zonotopes. The paper is organized as follows. In Section 2, we review the basic facts about zono-topes. In Section 3, the q-analog of the chromatic symmetric function (r) of a graph r is introduced. Theorem 1.1 is proved in Section 4. We present some examples and calculations in Section 5. 2 Zonotopes A zonotope Z = Z(v1,..., vm) is a convex polytope determined by a collection of vectors {v1,..., vm} in Rn as the Minkowski sum of line segments Z = [-Vi,Vi] +-----+ [-Vm,»m]. It is a projection of the m-cube [-1,1]m under the linear map t ^ At, t G [-1,1]m, where A = [v1 • • • vm] is an n x m-matrix whose columns are vectors v1,..., vm. The zonotope Z is symmetric about the origin and all its faces are translations of zonotopes. To a collection of vectors {v1,..., vm} is associated a central arrangement of hyperplanes H = {HV1,..., HVm}, where Hv denotes the hyperplane perpendicular to a vector v G Rn. The zonotope Z and the corresponding arrangement of hyperplanes H are closely related. In fact the associated fan of the arrangement H is the normal fan N(Z) of the zonotope Z (see [10, Theorem 7.16]). It follows that the face lattice of and the reverse face lattice of Z are isomorphic. In particular, vertices of Z correspond to regions of H and their total numbers coincide fo(Z )= r(H). (2.1) The faces of the zonotope Z are encoded by covectors of the oriented matroid M associated to the collection of vectors {v1,..., vm}. The covectors are sign vectors V* = {sign(v) G {+, -, 0}m | v G Rn}, ( +, (v,vj) > 0 where sign(v)j = < 0, (v, v4) =0 , i = 1,..., m. The face lattice of the zonotope Z [ -, (v,vi) < 0 is isomorphic to the lattice of covectors componentwise induced by +, - < 0 on V *. V. Grujic: Counting faces of graphical zonotopes 229 A special class of zonotopes is determined by simple graphs. To a connected graph r = (V, E), whose vertices are enumerated by integers V = {1,..., n}, are associated the graphical zonotope Zr = Z(ei - ej | i < j, {i,j} e E) and the graphical arrangement in Rn Hr = {Hei-ej | i 1 then the complementary set has k edges and Cn/F = Ck. Since a(Ck) = 2k - 2, k > 1, by formula (4.2), we obtain fn-k (Zen ) = (2k - 2)Q, 2 < k < n, which leads to the required formula. □ Specially, for n = 4 the resulting zonotope is the rhombic dodecahedron (see Figure 2). We have f (Ze4, q) = 14 + 24q + 12q2 + q3. Proposition 5.3. Let r = r1 Vv r2 be the wedge of two connected graphs r1 and r2 at the common vertex v. Then f (Zr,q) = f (Zn, q)f (Zr2,q). V. Grujic: Counting faces of graphical zonotopes 233 Proof. The graphical matroids of involving graphs are related by M(T) = M(Ti) © M(T2). For the sets of flats it holds F(T) = {Fi U F2 | Fi e F(r),i = 1, 2}. For F = F1 U F2 we have r/F = r1/F1 V [v] r2/F2, where [v] is the component of the vertex v in rv,F. Obviously a(r/F) = a(ri/Fi)a(r2/F2) and rk(F) = rk(Fi) + rk(F2). The proposition follows from formula (4.1). □ The formula for cubes in Example 5.1 (ii) follows from Proposition 5.3 since any tree is a consecutive wedge of edges and f (Ii, q) = q + 2. It also allows us to restrict ourselves only to biconnected graphs. For a biconnected graph r with a disconnecting pair of vertices {u, v} Whitney introduced the transformation called the twist around the pair {u, v}. This transformation does not have an affect on the graphical matroid M(r) [9]. Figure 3: Biconnected graphs related by twist transformation. Example 5.4. Figure 3 shows the pair of biconnected graphs on six vertices obtained one from another by the twist transformation. The corresponding zonotopes have the same f-polynomial f (Zn , q) = f (Zr2, q) = 126 + 348q + 358q2 + 164q3 + 30 q4 + q5. On the other hand their q-chromatic symmetric functions are different. One can check that corresponding coefficients by m3 i 3 are different [ms,i3(ri) = (11q2 +8q + 1) • 3!, [ms,i3(r) = (10q2 + 10q) • 3!. This shows that the q-analog of the chromatic symmetric function of a graph is not determined by the corresponding graphical matroid. By taking q = 0 we obtain that even the chromatic symmetric functions are different since [m3 i3]^(ri) = 6 and [m3 i3]^(r2) = 0. Let us now consider Stanley's example of nonisomorphic graphs with the same chromatic symmetric functions, see [7]. We find that the f -polynomials of the corresponding graphical zonotopes differ for those graphs. From these examples we conclude that chromatic properties of a graph and the f-vector of the corresponding graphical zonotope are not related. We have already noted that graphical zonotopes are generalized permutohedra. The h-polynomials of simple generalized permutohedra are determined in [5, Theorem 4.2]. The only simple graphical zonotopes are products of permutohedra [5, Proposition 5.2]. They are characterized by graphs whose biconnected components are complete subgraphs. Therefore Proposition 5.3 together with Example 5.1 (i) prove that the h-polynomial of a 234 Ars Math. Contemp. 13 (2017) 107-123 simple graphical zonotope is the product of Eulerian polynomials, the fact obtained in [5, Corollary 5.4]. Example 3.1 is of this sort and represents the hexagonal prism which is the product ZK3 x ZK2. References [1] M. Aguiar, N. Bergeron and F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), 1-30, doi:10.1112/s0010437x0500165x. [2] C. Benedetti, J. Hallam and J. Machacek, Combinatorial Hopf algebras of simplicial complexes, SIAMJ. Discrete Math. 30 (2016), 1737-1757, doi:10.1137/15m1038281. [3] B. Humpert and J. L. Martin, The incidence Hopf algebra of graphs, SIAM J. Discrete Math. 26 (2012), 555-570, doi:10.1137/110820075. [4] A. Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not. 2009 (2009), 1026-1106, doi:10.1093/imrn/rnn153. [5] A. Postnikov, V. Reiner and L. Williams, Faces of generalized permutohedra, Doc. Math. 13 (2008), 207-273, http://www.math.uiuc.edu/documenta/vol-13/10.html. [6] R. P. Stanley, Acyclic orientations of graphs, Discrete Math. 5 (1973), 171-178, doi:10.1016/ 0012-365x(73)90108-8. [7] R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Adv. Math. 111 (1995), 166-194, doi:10.1006/aima.1995.1020. [8] R. P. Stanley, An introduction to hyperplane arrangements, in: E. Miller, V. Reiner and B. Sturmfels (eds.), Geometric Combinatorics, American Mathematical Society, Providence, Rhode Island, volume 13 of IAS/Park City Mathematics Series, pp. 389-496, 2007. [9] H. Whitney, 2-isomorphic graphs, Amer. J. Math. 55 (1933), 245-254, doi:10.2307/2371127. [10] G. M. Ziegler, Lectures on Polytopes, volume 152 of Graduate Texts in Mathematics, SpringerVerlag, New York, 1995, doi:10.1007/978-1-4613-8431-1. ARS MATHEMATICA CONTEMPORANEA Author Guidelines Before submission Papers should be written in English, prepared in ETEX, and must be submitted as a PDF file. The title page of the submissions must contain: • Title. The title must be concise and informative. • Author names and affiliations. For each author add his/her affiliation which should include the full postal address and the country name. If avilable, specify the e-mail address of each author. Clearly indicate who is the corresponding author of the paper. • Abstract. A concise abstract is required. 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