ARS MATHEMATICA CONTEMPORANEA Volume 9, Number 1, Fall/Winter 2015, Pages 1-144 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 Is it time for electronic only? The speed at which the internet is changing the way we do research is fascinating. The role of libraries, books and journals is in constant transition. Many of the most important scientific works have been digitised. Our habits are changing too. Some of us still like to browse the paper version of books and journals, but in a majority of cases, when we need a copy of a mathematical paper, a pdf version is preferred over the paper copy. And information about new research no longer needs paper. In our journal, the on-line version is freely available to everyone, months in advance of the printed version. Our open access is neither gold nor green; it is diamond access: the reader does not pay, and the author does not pay. Accordingly, the title question seems to have an obvious positive answer. But is it really time for electronic only version of our journal? Well, not quite yet. There are some benefits to the printed version that should not be overlooked. For instance, support for the journal can be expressed by paying a subscription to the printed version. At scientific meetings, the journal can be promoted by exhibiting the printed copies. Distributed printed copies are still the best guarantee that the information presented in the papers will never be lost. In the era of a fast-growing market of predatory publishers of scientific internet journals of dubious quality, a look at the paper copy may be a distinguishing sign to a potential author. When a printed version comes out of a printshop, it marks a clear time-stamp for the current issue, which cannot then be altered. All of these arguments make us reluctant to abandon the paper copies of the journal for good. But we will still print as few copies as are really needed — at least for a few more years. Dragan Marušic and Tomaž Pisanski Editors In Chief Contents 4-Valent one-regular graphs of order 6p2 Mohsen Ghasemi, Pablo Spiga........................ 1 Partial product of graphs and Vizing's conjecture Ismael Gonzalez Yero............................. 19 Cycle construction and geodesic cycles with application to the hypercube Paul C. Kainen................................27 Comparing the irregularity and the total irregularity of graphs Darko Dimitrov, Riste Škrekovski......................45 Maximum genus, connectivity, and Nebesky's Theorem Dan Archdeacon, Michal Kotrbok, Roman Nedela, Martin Škoviera .... 51 The automorphism groups of non-edge-transitive rose window graphs Ted Dobson, Istvän Koväcs, Štefko MiklaviC.................63 Minimal covers of equivelar toroidal maps Kostiantyn Drach, Mark Mixer........................77 Group distance magic labeling of direct product of graphs Marcin Anholcer, Cichacz Sylwia, Iztok Peterin, Aleksandra Tepeh.....93 On spectral radius and energy of complete multipartite graphs Dragan Stevanovic, Ivan Gutman, Masood Ur Rehman...........109 Laceable knights Michael Dupuis, Stan Wagon.........................115 Matrices and their Kirchhoff graphs Joseph D. Fehribach..............................125 Volume 9, Number 1, Fall/Winter 2015, Pages 1-144 ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 1-18 4-valent graphs of order 6p2 admitting a group of automorphisms acting regularly on arcs Mohsen Ghasemi Department of Mathematics, Urmia University, Urmia 57135, Iran Pablo Spiga Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy Received 9 May 2012, accepted 5 February 2014, published online 7 May 2014 In this paper we classify the 4-valent graphs having 6p2 vertices, with p a prime, admitting a group of automorphisms acting regularly on arcs. As a corollary, we obtain the 4-valent one-regular graphs having 6p2 vertices. Keywords: One-regular graphs, 4-valent, Cayley graphs. Math. Subj. Class.: 20B25, 05C25 1 Introduction For a finite, simple and undirected graph X, we use V(X), A(X) and Aut(X) to denote its vertex set, arc set and automorphism group, respectively. The graph X is said to be B-vertex-transitive, respectively B-arc-transitive, if B is a subgroup of Aut(X) acting transitively on V(X), respectively A(X). When B = Aut(X), the prefix B in the above notation is omitted. Moreover, X is said to be one-regular if Aut(X ) acts regularly on A(X), that is, X is arc-transitive and | Aut(X)| = |A(X)|. Obviously, one-regular graphs are connected, and a graph of valency 2 is one-regular if and only if it is a cycle. The first example of a 3-valent one-regular graph was constructed by Frucht [10], with 432 vertices. Later on, a considerable amount of work has been done on 3-valent one-regular graphs as part of the more general problem dealing with the classification of the 3-valent arc-transitive graphs (see [5, 6, 7, 8, 19, 30]). Marusic and Pisanski [17] have fully classified the 3-valent one-regular Cayley graphs on a dihedral group, and Kwak et al. [14] have similarly classified those of valency 5. Moreover, more E-mail addresses: m.ghasemi@urmia.ac.ir (Mohsen Ghasemi), pablo.spiga@unimib.it (Pablo Spiga) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ Abstract recently, Feng and Li [9] have classified one-regular graphs of square-free order and of prime valency. Tetravalent one-regular graphs have also received considerable attention (see [1, 2, 15, 16, 24, 25, 28, 29]). In this paper we are concerned with the classification of the 4-valent one-regular graphs. We recall that such graphs are already classified when their orders are a prime or the product of two (not necessarily distinct) primes [3,21,22,26,29,28]. Moreover, for p and q primes, the classification of the 4-valent one-regular graphs of order 4p2 or 2pq is given in [4, 31]. In this context we prove the following. Theorem 1.1. Let p be a prime and let X be a 4-valent graph of order 6p2 admitting a group of automorphisms acting regularly on A(X ). Then one of the following holds: (i) X is isomorphic to C(2;3p2,1), C±1(p;6,2), Yp,±ъ Yp,±v3, Zp,±v-T or Zp,±v=3 (see Section 3 for the definition of these graphs); (ii) X is a Cayley graph over G with connection set S where (a) G = (x,y | xp = y6p = [x, y] = 1} and S = {y, y-1,xy, (xy)-1}, or (b) G = (x,y,z | xp = y3p = z2 = [x,y] = [x,z] = 1,yz = y-1} and S = {xz,x-1z,xeyz,x-eyz} (here e2 = —1 (mod p) and p = 1 (mod 4)), or (c) G = (x,y,z | xp = y3 = z2 = [x,y] = [x,z] = 1,yz = y-1} and S = {xz, x-1z, xyz, x-1yz}, or (d) G = (x,y,z | xp = y3 = z2 = [x,y] = [x,z] = 1,yz = y-1} and S = {xz, x-1z, x£yz, x-eyz} (here e2 = —1 (mod p2)), or (e) G = (x, y,z,t | xp = yp = z3 = t2 = [x, y] = [x, z] = [x, t] = [y, z] = [y, t] = 1, zl = z-1} and S = {xt, x-1t, yzt, y-1zt}; (iii) p G {2, 3, 5} and X is described in Section 6. The definition of the graphs in part (i) requires a fair amount of notation and terminology, so we do not include their description in this introductory section. Observe that if p < 7, then ^(X)| = 24, 54, 150 or 294. Since a complete census of the 4-valent arc-transitive graphs of order at most 640 has been recently obtained by Potocnik, Spiga and Verret [18, 19], for p g {2, 3, 5,7}, the 4-valent graphs of order 6p2 admitting a group of automorphisms acting regularly on A(X) can be downloaded (in magma format) from [18]. The proof of Theorem 1.1 is based on "normal quotient" techniques. This method is very powerful and allows to obtain results like Theorem 1.1 when the order of the graph has a prime factorization that is not too complicated (the multiplicity and the number of prime factors are both small). However there are two natural limits to this technique. First, as the order of the graph X becomes more complicated, the local properties of the quotient graph XN might not be strong enough to be lifted to the graph X we start with (to see a concrete example of this situation see "Case XP = O" in the proof of Theorem 1.1). Second, we believe that results like Theorem 1.1 are useful only when the list of graphs is not too long or too cumbersome to use. Therefore, although some of our arguments apply to graphs having order more complicated than 6p2 we do not pursue this classification here because of the natural complications describing each possible family. A direct application of Theorem 1.1 gives the following. Corollary 1.2. Let p be a prime and let X be a 4-valent one-regular graph X of order 6p2. Then one of the following holds: (i) X is isomorphic to Yp,± 1, Yp ±^3, Zp,±y—■ or Zp,±y-3 (see Section 3 for the defini- tion of these graphs); (ii) X is a Cayley graph over G with connection set S where (a) G = (x,y | xp = y6p = \x, y] = 1} and S = {y, y-1 ,xy, (xy)-1}, or (b) G = (x,y,z | xp = y3p = z2 = [x,y] = [x, z] = 1 ,yz = y-1} and S = {xz,x-1z,xeyz,x-eyz} (here e2 = —1 (mod p) and p = 1 (mod 4)), or (c) G = (x,y,z | xp = y3 = z2 = [x, y] = [x, z] = 1,yz = y-1} and {xz, x-1z, xyz, x-1yz}, or (d) G = (x,y,z | xp2 = y3 = z2 = [x, y] = [x, z] = 1,yz = y-1} and {xz,x-1z,xeyz,x-eyz} (here e2 = —1 (mod p2)); (iii) p G {2,3, 5} and X is described in Section 6. Observe that we are not claiming that every graph in Corollary 1.2 is one-regular. The structure of the paper is elementary: in Section 2 we introduce the notation and some basic results that we will need for our proof of Theorem 1.1. Then in Sections 3 and 4 we present some graphs revelant to our investigation. In Section 5 we prove Theorem 1.1 and Corollary 1.2. In Section 6 we give the graphs in part (iii) of Theorem 1.1, and in part (iii) of Corollary 1.2. Acknowledgements. A toast to our friend Primož Potocnik for being constructively critical. 2 Preliminaries In this section, we introduce some notations and definitions as well as some preliminary results which will be used later in the paper. Let X be a connected vertex-transitive graph, and let B < Aut(X ) be vertex-transitive on X. Suppose that there is some group N such that 1 = N < B, and N is intransitive in its action on V(X). The normal quotient XN is the graph whose vertices are the orbits of N on V(X), with an edge between two distinct vertices vN and wN in V(XN), if and only if there is an edge of X between v0 and w0, for some v0 G vN and some w0 G wN. Normal quotients were introduced by Praeger in [20] and they turned out to be an invaluable tool for the classification of certain families of vertex-transitive graphs. In fact, our proof of Theorem 1.1 heavily relies on normal quotient techniques. For a positive integer n, we denote by Zn the cyclic group of order n as well as the ring of integers modulo n, by Z*n the invertible elements of Zn, by D2n the dihedral group of order 2n, and by Cn and Kn the cycle and the complete graph of order n, respectively. For a group G and a subset S of G with 1 G S and S = S-1, the Cayley graph Cay(G, S) on G with connection set S is defined to have vertex set G and edge set {{g,sg} | g g G, s g S}. For the benefit of the reader, we report here [12, Theorems 1.1 and 1.2] and [11, Theorem 1.1], respectively. Theorem 2.1 ([12, Theorem 1.1]). Let X be a connected B-arc-transitive 4-valent graph, and let N be a minimal normal p-subgroup of B with orbits on V(X) of size ps. Let K denote the kernel of the action of B on N orbits and let a be a vertex of X. If the quotient XN is a cycle of length r > 3, then one of the following holds: (a) p = 2 and X = C(2; r,s); (b) p is odd and, if \Ka\ = 2s, then X = C±1(p; st, s) or X = C±e(p; 2st, s) for some t > 1. Theorem 2.2 ([12, Theorem 1.2]). Let X be a connected B-arc-transitive 4-valent graph, and let N = Zp (p an odd prime) be a minimal normal subgroup of B with orbits on V (X ) of size ps. Let K denote the kernel of the action of B on N-orbits and let a be a vertex of X. If the quotient XN is a cycle of length r > 3, then one of the following holds: (a) s = 1 or 2, Ka = Zs2 and X = C±1(p; st, s) or X = C±e(p; 2st, s) for some t > 1; (b) s = 2, Ka = Z2 and X = C±1(p;2t, 2), or X belongs to one of two families described in [12, Lemmas 8.4 and 8.7]. (The graphs C(2; r, s), C±1(p; st, s), C±e(p; 2st, s) and the graphs in [12, Lemma 8.4 and 8.7] are define in Section 3.) Theorem 2.3 ([11, Theorem 1.1]). Let X be a connected 4-valent B-arc-transitive graph. For each normal subgroup N of B, one of the following holds: (a) N is transitive on V(X); (b) X is bipartite and N acts transitively on each part of the bipartition; (c) N has r > 3 orbits on V(X), the quotient graph XN is a cycle of length r, and B induces the full automorphism group D2r on XN; (d) N has r > 5 orbits on V(X), N acts semiregularly on V(X), the quotient graph XN is a connected 4-valent B/N-symmetric graph, and X is a B-normal cover of XN. 3 Some families of graphs In this section we present some of the graphs relevant to Theorem 1.1 and Corollary 1.2. All of these graphs were introduced in the pivotal paper [12] of Gardiner and Praeger. (Here we do not include the description of all the graphs in [12], but only those that are closely related to our investigation.) 3.1 The graphs C(2; r, 1) The graph C(2; r, 1) is the lexicographic product of a cycle of length r and an edgeless graph on 2 vertices. In other words, V(C(2; r, 1)) = Z2 x Zr with (u, i) being adjacent to (v,j) if and only if i - j e {-1,1}. From [12, Definition 2.1], it follows that C(2; r, 1) is not one-regular. 3.2 The graphs C(p; st, s) (We will be only interested to the case s e {1,2}.) The graph C±1(p; st, s) has vertex set Zp x Zst. To describe the adjacencies write every vertex of C±1(p; st, s) as v = ((xo,. .., xi-1, xi, xi+1, .. ., xs-1),ms + i), with x0,..., xs-1 e Zs, 0 < i < s and 0 < m 5, we have Aut(Yp,u ) = Bu. Next we define Yp ±^3. Suppose p > 5 and let u g Zp be a square root of 3 (mod p). (Observe that, from the law of quadratic reciprocity, for this example to exist we need p = ±1 mod 12.) Let N = {a0, a1} be an elementary abelian p-group of order p2 and let K be the group with presentation K = {a, t, w | a12 = т2 = w2 = [a, w] = [т, w] = 1, a6 = = a-1}. Clearly, K = {a, т} = D24. We let K act on N via a1 1 a1 a0 a0 Set Bu = N xK and Hu = {w,t}. As for Yp,±1, the graph Yp,u is defined by Cos (Bu, Hu, a-1a0). From Proposition 3.1, it follows that Yp,u is a Bu-arc-transitive graph of valency 4 with 6p2 vertices and Bu acts regularly on A(Yp u). Moreover, from [12, Section 8], we have Ypyg = Yp_^s and Aut(Yp,u) = Bu. 3.5 The graphs arising from [12, Lemma 8.7]: 3 and 1 As for Section 3.4, the graphs described in [12, Section 8 and Lemma 8.7] are very general, here we present only the graphs relevant to the scope of this article. We start by defining Suppose p > 5 and let u g Zp be a square root of -3 (mod p). (Observe that, from the law of quadratic reciprocity, for this example to exist we need p = 1 mod 6.) Let N = {a0, a1} be an elementary abelian p-group of order p2 and let K be the group with presentation K = {a, т, w | a6 = т4 [a, w] = [т, w] = 1, т2 t 2-1 w, a = т a- }. We let K act on N via a1 u a0a1 a a1 -1 0 1 Set Bu = N x K and Hu = {т}. Now the graph Zp,u is defined by Cos(Bu, Hu, a-1a0). From Proposition 3.1, it follows that Zp u is a Bu-arc-transitive graph of valency 4 with 6p2 vertices and Bu acts regularly on A(Zp,u). Moreover, from [12, Section 8], we have Z Zp _y=3 and Aut(Zp,u) Bu ? a= a 0 ? w a a =a a 1 1 1 w, a 1 aT = w a a 0 1 a? = u aT = a 1 1 1 2 w ? T w a= a= a 0 0 1 w a= a a =a 1 1 Finally, we define Zp,±y-T. Suppose p > 5 and let u g Zp be a square root of -1 (mod p). (Observe that, from the law of quadratic reciprocity, for this example to exist we need p = 1 mod 4.) Let N = (a0, a^ be an elementary abelian p-group of order p2 and let K be the group with presentation K = (a, т, w | a12 = T4 = w2 = [a, w] = [т, w] = 1, a6 = т2 = w, aT = a-1}. We let K act on N via ag" = ai a0 = ai aW = a-1 ag = aoa" , a^ = a-1 , aW = a-1 . Set Bu = N x K and Hu = (т}. As for Zp,±y-3, the graph Zp,u is defined by Cos(Bu, Hu, a-1a0). From Proposition 3.1, it follows that Zp,u is a Bu-arc-transitive graph of valency 4 with 6p2 vertices and Bu acts regularly on A(Zp,u). Moreover, from [12, Section 8], we have Zp ^-I = Zp,_y-I and Aut(Zp,u) = Bu. 4 Graphs for Theorem 1.1 (ii) In this section we describe the examples introduced in Theorem 1.1 (ii) (and their relation to the graphs in Section 3). For simplicity here we assume that p > 5. The graphs described in Sections 4.1 and 4.6 were already introduced with much more information and details in [28], here we include yet again their construction for making our note self-contained. 4.1 4-Valent normal Cayley graphs over Zp X Z6p with p > 5 prime We describe the connected normal 4-valent arc-transitive Cayley graphs over the group Zp X Z6p. Let p be a prime and let G be the group given by generators and relations G = (x,y | xp = y6p = [x,y] = 1}. Let S be a subset of G and assume that X = Cay(G, S) is connected, normal, 4-valent and arc-transitive. Since X is normal and arc-transitive and since G is abelian of exponent 6p, we see that S consists of elements of order 6p. Denote by S6p the elements of G of order 6p. Therefore S CS6p = {xV | a g Zp,b G Z*p}. It is clear that Aut(G) acts transitively on S6p by conjugation. In particular, replacing 5 by a suitable Aut(G)-conjugate, we may assume that y G S. Therefore S = {y, y , xu yv ,x-uy-v }, for some u G Z* and for some v G Z6p. Let B = {^ G Aut(G) | yv = y}. Given ^ G B, we have : fx ^ x°y6b ^ ' 1 y ^ y with a, b G Zp and a = 0. Note that every invertible element of Z6p is of the form 1 + 6b or —1 + 6b, for some b g Zp. Therefore, we may choose a, b G Zp with (xy)v = xuyv or (xy 1)v = xuyv. Thus, replacing S by a suitable B-conjugate, we may assume that either xy e S or xy-1 e S, that is, Let ( be the automorphism of G with xv = x and yv = y 1. Clearly, ( maps the first possibility for S onto the second. Therefore, we may assume that The graph X is in Theorem 1.1 (ii) (a). Also, using [12, Definition 2.2], we see that X is isomorphic to C±1(p; 6p, 1). 4.2 4-Valent normal Cayley graphs over Zp x D6p with p > 5 prime We describe the connected normal 4-valent arc-transitive Cayley graphs over the group Zp X D6p. Let p be a prime and let G be the group given by generators and relations Let S be a subset of G and assume that X = Cay(G, S) is connected, normal, 4-valent and arc-transitive. Since X is normal and arc-transitive, every element of S has the same order. The elements of G of odd order lie in (x, y) and the involutions of G lie in (y, z). Since X is connected, G = (S) and hence S consists of elements of order 2p. Denote by S2p the elements of G of order 2p. Therefore S с S2p = {xaybz | b e Z3p, a e Z;}. We now consider the action of the automorphism group Aut(G) of G on S2p. Let ( e Aut(G). Clearly, ( is uniquely determined by the images of x, y and z. By considering the element orders of G, we need to have Since [x, z] = 1, the element xv needs to commute with zv and so, with a direct computation, we see that 3b = 0. Also, as yz = y-1, we have (yv)zV = (yv)-1, but this happens only for c = 0. Summing up, This shows that all elements of S2p are Aut(G)-conjugate to xz. Therefore, replacing S by a suitable conjugate under Aut(G), we may assume that xz e S. In particular, S = {xz, x-1z, xuyvz, x-uyvz}, for some u, v. As G = (S) < (x, z, yv), we have that S = {y, y 1,xy, x 1y 1}, or S = {y, y-1, xy-1, x-1y}. S = {y,y 1,xy,x 1y 1}. G = (x, y, z | xp = y3p = z2 = [x, y] = [x, z] = 1, yz = y 1). x ^ xay3b c d y ^ xcyd z ^ yez. (4.1) Let B = {( e Aut(G) | (xz)v = xz}. From (4.1), we see that ф e B only if x^ = x and z^ = z. In particular, replacing S by a suitable B-conjugate, we may assume that xuyz e S, that is, v = 1. Thus S = {xz, x-1z, xuyz, x-uyz}. Let ( e Aut(G) with Sv = S and (xz)v = xuyz (recall that such an automorphism exists because X is a normal arc-transitive Cayley graph over G). From (4.1), we have xv = xu, zv = yz and yv = yd (for some d coprime to 3p). Now, (xuyz)v = xu yd+1 z e S. If xu yd+1z e {xuyz, x-uyz}, then d = 0, contradicting the fact that d is coprime to 3p. Therefore xu yd+1z e {xz, x-1z} and u2 = ±1. Thus we have one of the following two possibilities for S: S = {xz,x-1z,xyz,x-1yz}, or S = {xz,x-1z,xeyz,x-eyz}, where e2 = —1 (note that in the second case p = 1 (mod 4)). Now we focus our attention to the first possibility for S. Take {x ^ x y ^ y-1 z ^ yz. Since ф fixes set-wise S, we have G x (ф) < Aut(X). We see that (z-'ф)2 = zфzф = z(фzф) = zz^ = zyz = yz = y-1 and so zф has order 6p. Moreover xz^ = (xz= x^ = x and x commutes with z—. Therefore (x, zф) = Zp x Z6p. It is immediate to see that (x, zф) acts regularly on the vertices of X and so X is a Cayley graph over Zp x Z6p. In particular, from the discussion in Section 4.1 (or with a direct computation) we obtain that X is isomorphic to the graph in Theorem 1.1 (ii) (a). Therefore we may assume that p = 1 (mod 4) and that S = {xz, x-1z, xeyz, x-eyz} where e2 = —1. The graph X is in Theorem 1.1 (ii) (b). Also, using [12, Definition 2.3] (or Section 3.3), we see that X is isomorphic to C±E(p; 6p, 1). 4.3 4-Valent normal Cayley graphs over Zp X Zp X D 6 with p ^ 5 prime We proceed as in the previous two examples. Let p be a prime and let G be the group given by generators and relations G = (x, y,z,t | xp = yp = z3 = t2 = [x, y] = [x, z] = [x, t] = [y, z] = [y, t] = 1, z* = z-1). Let S be a subset of G and assume that X = Cay(G, S) is connected, normal, 4-valent and arc-transitive. As the elements of G of odd order lie in (x, y, z) and the involutions of G lie in (z, t), we must have that S consists of elements of order 2p. Since (x, y) and (z,t) are characteristic subgroups of G, we have Aut(G) = Aut((x,y)) x Aut((z,t)) = GL2 (p) x D6. It follows easily from this description of Aut(G) and the connectivity of X that we may assume that S = {xt, x-1t, yzt, y-1zt}. Thus we obtain the graphs in Theorem 1.1 (ii) (e). We show that X is not one-regular and hence it is not relevant for Corollary 1.2. Take ф : x ^ x 1 У ^ У z ^ z t t. Clearly, ф defines an automorphism of G that fixes set-wise S. Since ф fixes the neighbour yzt of 1 and maps xt to x-1t, we see that Aut(X) is not regular on A(X). 4.4 4-Valent normal Cayley graphs over Zßp X D2p with p ^ 5 prime Let G be the group given by generators and relations G = (x,y,z | x3p = yp = z2 = [x,y] = [x,z] = 1,yz = y-1). Let S be a subset of G and assume that X = Cay(G, S) is connected, normal, 4-valent and arc-transitive. As the elements of G of odd order lie in (x, y), the involutions of G lie in (y, z) and the elements of order 2p lie in (x3, y, z), we must have that S consists of elements of order 6p. Denote by S6p the elements of G of order 6p. Therefore S С Sep = jxVz | a G Z3p, b G Zp}. Arguing as in the previous examples, we see that Aut(G) acts transitively on S6p and hence we may assume that xz G S. In particular, S = {xz, x-1z, xuyvz, x-uyvz}, for some u G Z3p and some v G Zp. Let B = {<£> G Aut(G) | (xz)v = xz}. Clearly, if (xz)v = xz, then xv = x and zv = z because z = (xz)3p and x2 = (xz)2. Using this observation, it is easy to see that the elements ^ G B are of the form ^ : < y ^ x3ayb for some a, b G Zp with b = 0. Therefore, we may choose a and b with (x"yvz)v = xyz or (x"yvz)v = x-1yz. Therefore (as usual), replacing S by a suitable B-conjugate, we may assume that S = {xz, x-1z, xyz, x-1yz}. Take {x ^ x y ^ y-1 z ^ yz. Clearly, ф defines an automorphism of G. Since ф fixes set-wise S, we have G x (ф) < Aut(X ). We see that x x z z ^ф)2 = zфzф = z(фzф) = zz^ = zyz = yz = y 1 and so z-ф has order 2p. Moreover xz^ = (xz= x^ = x and x commutes with х'ф. Therefore (x, z^} = Z3p x Z2p = Zp x Z6p. It is immediate to see that (x, z^} acts regularly on the vertices of X and so X is a Cayley graph over Zp x Z6p. In particular, from the discussion in Section 4.1 (or with a direct computation) we obtain that X is isomorphic to the graph in Theorem 1.1 (ii) (a). 4.5 4-Valent normal Cayley graphs over Zp2 x D6 with p > 5 prime Let G be the group given by generators and relations G = (x,y,z | xp = y3 = z2 = [x, y] = [x, z] = 1,yz = y-1}. Let S be a subset of G and assume that X = Cay(G, S) is connected, normal, 4-valent and arc-transitive. A moment's thought gives that S consists of elements of order 2p2. Denote by S2p2 the elements of G of order 2p2. Therefore S С S2p2 = {xaybz | a e Zp2, b e Z3}. Since (x} and (y, z} are characteristic subgroups of G, we have Aut(G) = Aut((x}) x Aut((y, z}) = Z*2 x D6 and Aut(G) acts transitively on S2p2. Hence we may assume that xz e S. In particular, S = {xz, x-1z, x"yvz, x-uyvz}, for some u e Zp2 and some v e Z3. Replacing S by a suitable Aut(G)-conjugate, we may assume that v = 1, that is, S = {xz, x-1z, x"yz, x-uyz}. Let ^ e Aut(G) with Sv = S and (xz)v = x"yz (recall that such an automorphism exists because X is a normal arc-transitive Cayley graph over G). From the description of Aut(G), we have xv = xu and zv = yz and yv = yd (for some d e Z3). Now, (x"yz)v = xu yd+1z e S. As d = 0, we obtain d = —1 and u2 = ±1. Thus we have one of the following two possibilities for S: S = {xz, x-1z, xyz, x-1yz}, or S = {xz, x-1z, xeyz, x-eyz}, where e2 = —1 (note that in the second case p = 1 (mod 4)). In particular, we obtain that X is isomorphic to the graph in Theorem 1.1 (ii) (c) or (d). 4.6 4-Valent normal Cayley graphs over Zp2 x Z6 with p > 5 This case is by far the easiest to deal with and we leave it to the conscious reader. There is only one graph arising, namely the graph in Theorem 1.1 (ii) (c). 5 Proof of Theorem 1.1 and Corollary 1.2 We start with some technical preliminary lemmas that will be useful in the proof of Theorem 1.1. Lemma 5.1. Let p be a prime with p > 5 and let N be a normal subgroup of G with |N| = p and G/N = Zp x Z6 or G/N = Zp x D6. Then G is isomorphic to one of the following groups (i) Zp2 X Z6 or Zp2 X D6; or (ii) Zp X (Zp X Z6) or Zp X (Zp X D6). Proof. Let P be a Sylow p-subgroup of G. Since G is soluble, G contains a subgroup Q with |Q| = 6. As G/N ^ Zp x Ze or G/N ^ Zp x D6, we have that QN/N = Q (that is, Q = Z6 or Q = D6) and that Q centralizes P/N. If P is cyclic, then Q centralizes P by [13, Theorem 1.4]. So G = P x Q and part (i) follows. Suppose that P is an elementary abelian p-group. The action of Q by conjugation on P endows P of a structure of an ZpQ-module. As Q has order coprirne to p, the ZpQ-module N is completely reducible, that is, P = N x N', for some normal subgroup N' of G of size p. As Q centralizes P/N, we see that Q centralizes N'. Thus G = N' x (N x Q) and part (ii) follows. □ Lemma 5.2. Let p be a prime with p > 7 and let X be a connected normal 4-valent arc-transitive Cayley graph over one of the groups G in Lemma 5.1 (ii). Let P be a Sylow p-subgroup of G. Assume that P < G and that Xp = Ce. Then either G = Zp X Z6p, or G = Z3p X D2p, or G = Zp X D6p, or G = Zp x Zp x D6. Proof. From Lemma 5.1 (ii), we have G = (ж) x ((y) x (z, t)) with (ж, y) = Zp x Zp, |z| = 3 and |t| = 2. Moreover, [z,t] = 1 if G = Zp x (Zp x Z6) and z4 = z-1 if G = Zp X (Zp X D6). If z centralizes y, then G = (ж) x ((yz) x (t)). In particular, G = Zp X Z6p if t centralizes both y and z, G — Zp X D6p if t inverts both y and z, G = Z3p X D2p if t centralizes z and inverts y, and G = Zp x Zp x D6 if t inverts z and centralizes y. In remains to consider the case that z does not centralize y. We show that this case actually does not arise (here we use the fact that G admits a normal Cayley graph). Note that the automorphism group of Zp is cyclic and hence D6 cannot act faithfully as a group of automorphisms on Zp. Since we are assuming that z does not centralize y, we must have that G = Zp x (Zp x Z6), that is, [z, t] = 1. Let S be a subset of G with X = Cay(G, S). Write A = Aut(X). Since (ж) is a characteristic subgroup of G and since G < A, we obtain that (ж) < A and Y = X^) is a normal quotient having 6p vertices. In particular, Y has valency 2 or 4. Let K be the kernel of the action of A on V(Y) and assume that Y is a cycle. In particular, A/K = D12p. Now D12p contains exactly two regular subgroups, one isomorphic to Z6p and the other to D6p. Therefore GK G G ■K = GnK = (ж) = (y) X (z,t) is isomorphic either to Z6p or to D6p, contradicting the fact that z does not centralize y. Thus Y is a 4-valent Cayley graph over GK/K = (y) x (z, t) and K = (ж). Now, P/K has order p and is therefore a minimal normal subgroup of A/K. Also, YP/K = XP = C6 and so we are in the position to apply Theorem 2.1 (b) to A/K, P/K and Y. Hence Y = C±1(p; 6,1) or Y = C±e(p; 6,1). From Sections 3.2 and 3.3 we see that Y is a Cayley graph over Zp x Z6 or over Zp x D6. However, (y) x (z, t) is isomorphic to neither of these groups, a contradiction. □ In what follows we denote by O the graph K6 - 6K2, that is, K6 with a perfect matching removed. Clearly, O is connected, 4-valent and |V(O)| = 6. We refer to O as the Octahedral graph. Proof of Theorem 1.1. We first consider the case that p > 11. Let B be a subgroup of Aut(X) acting regularly on A(X), let Bv be the stabilizer in B of the vertex v g V(X) and let P be a Sylow p-subgroup of B. We show that P is normal in B. Since |B| = |A(X)| = 24p2, Sylow's theorems show that the number of Sylow p-subgroups of B is equal to |B : NB (P)| = 1 + kp, for some k > 0. If k = 0, then P is normal in B and thus we may assume that k > 1. Now, 1 + kp divides 24 and this is possible if and only if k =1 and p = 23, or k = 1 and p = 11. Suppose that k = 1 and p = 23. Now |B : NB (P)| = 24. So NB (P) = P and CB (P) = NB (P). Therefore, by the Burnside's p-complement theorem [27, page 76], we see that B has a normal subgroup N of order 24. In particular, P acts by conjugation as a group of automorphisms on N. As a group of order 24 does not admit non-trivial automorphisms of order 23, we see that P centralizes B. Thus B = N X P and P is normal in B. Finally, suppose that k =1 and p =11. Now, |B : NB(P)| = 12. Consider the permutation group B induced by the action of B on the cosets of NB (P). Now, B has degree 12 and has order divisible by 11 because (by hypothesis) P is not normal in B. Therefore B is a 2-transitive group whose order divides 24 • 112. A quick inspection on the list of 2-transitive groups of degree 12 in magma shows that this is impossible. This final contradiction gives that P is normal in B. As usual, we denote by XP the normal quotient of X via P. The rest of the proof is a case-by-case analysis depending upon the structure of the normal quotient XP .At the end of the proof of each case we use the symbol ■ to mean that the theorem is proved in the case under consideration. As |Bv | = 4, we have Bv n P = 1 and P acts semiregularly on V(X). Thus the orbits of P on V(X) have size p2 and |V(XP)| = 6. In particular, XP is either the cycle C6 or the octahedral graph O (depending on whether XP has valency 2 or 4). Case A: XP = O. Fix v a vertex of X and let K be the kernel of the action of B on V (XP ). As XP has valency 4, we obtain |B/K| = 24 and K = P. Now, the automorphism group W of O is isomorphic to the wreath product Z2 l Sym(3), which has order 48. Therefore B/P is isomorphic to a subgroup of index 2 in W. Labelling the vertices of O as {1, 2, 3,4,5,6} (so that {{1,2}, {3,4}, {5, 6}} is the system of imprimitivity for W), we may assume that W = ((1, 2), (3, 4), (5, 6), (1, 3)(2,4), (1, 5)(2, 6)). The group W has exactly three subgroups of index 2. Namely, Wi = ((1, 2)(3,4), (1, 2)(5, 6), (1, 3)(2,4), (1, 5)(2, 6)), W2 = ((1, 2), (3, 4), (5, 6), (1, 3, 5)(2, 4, 6)), W3 = ((1, 2)(3,4), (1, 2)(5, 6), (1, 3, 5)(2,4, 6), (1, 2)(3, 6)(4, 5)). The group B/P acts by conjugation as a group of automorphisms on P. So, if B/P = Wj, then Wj admits an action as a group of automorphisms on P. Assume that B/P acts faithfully on P, that is, CB (P) = P. In particular, Wj admits a faithful irreducible action on P. Suppose that P is cyclic. Then Aut(P) is cyclic and so B/P is cyclic. However, W1, W2 and W3 are not cyclic, a contradiction. Thus P is an elementary abelian p-group and Aut(P) = GL2(p) (the group of 2 x 2 invertible matrices). It is clear that the group SL2(p) has index 2 in GL2(p) and that SL2(p) contains a unique element of order 2. This shows that Wj has a normal subgroup T with Wj/T cyclic and with T containing at most one involution. A direct inspection on W\, W2 and W3 shows that such a normal subgroup T does not exist. Therefore B/P does not act faithfully on P and CB (P) > P. Another direct inspection on W1, W2 and W3 shows that Wi and W3 contain a unique minimal normal subgroup (namely, ((1,2)(3,4), (1,2)(5, 6)}, which has order 4) and W2 contains exactly two minimal normal subgroups (((1, 2)(3,4)(5,6)} having size 2 and ((1,2)(3,4), (1,2)(5, 6)} having size 4). Therefore, CB(P) contains a minimal normal subgroup Q with |Q| = 2 or |Q| = 4. Suppose that |Q| = 2 (and so B/P = W2). Now, W2/((1,2)(3,4)(5, 6)} = Alt(4) the alternating group on 4 letters. Arguing as in the previous paragraph, we see that Alt(4) cannot act faithfully on P. Therefore | CB (P) : P| > 4 and CB (P) contains a minimal normal subgroup R with |R| = 4. So, replacing Q by R if necessary, we may assume that |Q| = 4. Since Q is characteristic in CB (P), we get that Q is normal in B. As 4 does not divide | V(X) |, we get that |Qv | = 2 and the Q-orbits have size 2. So, the quotient graph Xq is a cycle of length 3p2. Now from Theorem 2.1 (a), we obtain X = C(2; 3p2,1) and X is as in Theorem 1.1 (i). ■ For the remainder of the proof we may assume that XP = C6. Case B: P is a minimal normal subgroup of B. Fix v a vertex of X and let K be the kernel of the action of B on V(XP). As |B| = 24p2 and as XP has valency 2, we have B/K = D12 and |Kv| = 2. So, we see that Theorem 2.2 (b) applies (with s = 2), and X is isomorphic to C±1(p; 6, 2) or to one of the graphs defined in [12, Lemmas 8.4 and 8.7]. From Sections 3.2, 3.4 and 3.5, Theorem 1.1 (i) holds. ■ For the remainder of the proof we may assume that B has a minimal normal subgroup N with N < P and |N | = p. Now by Theorem 2.3 the normal quotient XN is either C6p or a 4-valent graph. Case C: Xn = C6p. Let K be the kernel of the action of B on V(XN). As |B| = 24p2 and XN is 2-valent, we have B/K = D12p and |Kv | = 2. So, we are in the position to apply Theorem 2.1 (b) (with s = 1) and thus X is isomorphic to either C±1(p; 6p, 1) or to C±e(p; 6p, 1). From Sections 4.1 and 4.2, we get that in the first case Theorem 1.1 part (ii) (a) holds and in the second case Theorem 1.1 part (ii) (b) holds. ■ For the remainder of the proof we may assume that XN is a 4-valent graph. So, X is a regular cover of XN. Denote by K the kernel of the action of B on V(XP ) and recall that |K | = 2p2 because XP = C6. Also, note that the normal quotient (XN )P/N is isomorphic to XP. Now P/N is a minimal normal subgroup of B/N with orbits of size p. The kernel of the action of B/N on XP is K/N and |KvP/P| = 2. Therefore Theorem 2.1 (b) applies (with s = 1 and with B replaced by B/N), and so XN = C±1(p; 6,1) or XN = C±e(p; 6,1). From Sections 3.2 and 3.3, we see that C±1(p; 6,1) is anormal Cayley graph over Zp X Z6 and that C±e(p; 6,1) is a normal Cayley graph over Zp x D6. Let G/N be the normal subgroup of B/N acting regularly on V(XN), with G/N = Zp X Z6 or with G/N = Zp x D6. Clearly, G acts regularly on V(X) and thus X is a normal Cayley graph over G. From Lemma 5.1, we see that G is isomorphic either to Zp2 X Z6, or Zp2 X D6, or Zp X (Zp x Z6), or Zp x (Zp x D6). If G is isomorphic to Zp X (Zp x Z6), or Zp X (Zp x D6), then from Lemma 5.2 G is isomorphic to Zp X Z6p, or Zp X D6p, or Zp X Zp X D6, or Z3p X D2p. Now the proof follows from Sections 4.1, 4.2,4.3 and 4.4. If G is isomorphic either to Zp2 x Z6 or Zp2 x D6, then the proof follows from Sections 4.5 and 4.6. It remains to consider the case that p < 7. When p = 7, we see from [18, 19] that there are seven 4-valent arc-transitive graphs on 6p2 vertices. A computer computation shows that six of these graphs are isomorphic to one of the graphs defined in Theorem 1.1 (i) and (ii), and the seventh does not admit a group of automorphisms acting regularly on arcs. When p = 5, we see from [18, 19] that there are ten 4-valent arc-transitive graphs on 6p2 vertices and they all admit a group of automorphisms acting regularly on arcs. It is a computer computation checking that eight of these graphs are isomorphic to one of the graphs defined in Theorem 1.1 (i) and (ii), and the remaining two are given in Section 6. When p = 3, we see from [18, 19] that there are five 4-valent arc-transitive graphs on 6p2 vertices and they all admit a group of automorphisms acting regularly on arcs. It is a computer computation checking that three of these graphs are isomorphic to one of the graphs defined in Theorem 1.1 (i) and (ii), and the remaining two are given in Section 6. When p = 2, the result follows again with a straightforward computation. □ Proof of Corollary 1.2. Observe that from Sections 3.1,3.2 and 4.3 the graphs C (2; 3p2,1), C±1 (p; 6, 2) and the graphs in Theorem 1.1 (ii) (e) are not one-regular. Now the result follows immediately from Theorem 1.1. □ 6 Description of the graphs in Theorem 1.1 part (iii) and Corollary 1.2 (iii) New now describe the exceptional graphs in Theorem 1.1 (iii) and Corollary 1.2 (iii). Case p = 2. (i) X = Cay ((ж), {x, x-1, x5, x-5}) where (x) is a cyclic group of order 24. Now, X is a normal one-regular Cayley graph. (ii) X = Cay((x), {x, x-1, x7, x-7}) where (x) is a cyclic group of order 24. Now, X is a normal one-regular Cayley graph. (iii) X = Cay(SL(2, 3), S) and with connection set Now, X is a normal one-regular Cayley graph. (v) X = Cay(G, S) with G = ((1,2,3), (1,2), (4, 5, 6,7)) = D6 x Z4 and S = {(1, 3)(4, 6)(5, 7), (1, 2, 3)(4, 5, 6, 7), (2, 3)(4, 6)(5, 7), (1, 3, 2)(4, 7, 6, 5)}. Now, X is neither a normal Cayley graph nor one-regular. Case p = 3. Now, X is a normal one-regular Cayley graph. (iv) X = Cay(SL(2, 3), S) and with connection set (i) X is the Cayley graph Cay(G, S), where G is the subgroup of GL(3, 3) generated by -10 0 0 1 0 0 0 1 1 1 0 010 001 100 0 1 1 0 0 1 and with connection set -1 -1 -1 0 1 -1 001 -1 -1 1 \ 0 1 1 I 0 0 1 J ( -1 1 1 \ ( -1 1 -1 I 0 1 -1 I , I -0 11 V 0 0 1 0 01 Now, it is a computation to verify that X is not one-regular. (ii) X is the Cayley graph Cay(G, S), where G is the subgroup of GL(3,3) generated by -1 0 0 0 1 0 0 0 1 1 1 0 010 001 100 0 1 1 0 0 1 and with connection set -1 0 0 0 1 0 0 0 1 -1 1 0 0 1 0I , 0 1 ( -1 -1 \ ( -1 -1 0 I 0 1 -1 I , I -0 1 -1 V 0 0 1 0 01 Now, it is a computation to verify that X is one-regular. Case p = 5. (i) X is the coset graph Cos(G, H, g) where G = ((1, 2, 3,4, 5), (2, 5)(3,4), (6, 7, 8, 9,10), (8, 9,10)) = D5 x Alt(5), H = ((1,5)(2,4)(6,9)(8,10), (6,10)(8, 9)) and g = (1,3)(4,5)(6,10)(7, 8). Now, | Aut(X ) | = 1200 and hence X is not one-regular. 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Zhou and Y.-Q. Feng, Tetravalent one-regular graphs of order 2pq, J. Algebraic Combin. 29 (2009), 457-471. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 19-25 Partial product of graphs and Vizing's conjecture Ismael Gonzalez Yero Departamento de Matemàticas, Escuela Politècnica Superior de Algeciras Universidad de Cadiz, Av. Ramon Puyol s/n, 11202 Algeciras, Spain Received 9 December 2012, accepted 19 December 2013, published online 5 June 2014 Abstract Let G and H be two graphs with vertex sets V1 = {u1,...,uni} and V2 = {v1,...,vn2}, respectively. If S с V2, then the partial Cartesian product of G and H with respect to S is the graph GDSH = (V, E), where V = Vi x V2 and two vertices (uj, vj) and (uk, vl) are adjacent in GDSH if and only if either (u = uk and vj ~ vl) or (u ~ uk and vj = vl g S). If A с V1 and B с V2, then the restricted partial strong product of G and H with respect to A and B is the graph GA H = (V, E), where V = V1 x V2 and two vertices (uj, vj) and (uk,vl) are adjacent in GA H if and only if either (uj = uk and vj ~ vl) or (uj ~ uk and vj = vl) or (uj g A, uk G A, vj g B, vl G B, uj ~ uk and vj ~ vl) or (uj G A, uk G A, vj GB, vl G B, uj ~ uk and vj ~ vl). In this article we obtain Vizing-like results for the domination number and the independence domination number of the partial Cartesian product of graphs. Moreover we study the domination number of the restricted partial strong product of graphs. Keywords: Domination, partial product of graphs, Cartesian product graph, strong product graph, Vizing's conjecture. Math. Subj. Class.: 05C69, 05C70, 05C76 1 Introduction Vizing's conjecture [7] is perhaps one of the most popular open problems related to domination in graphs. It states that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. A high quantity of approaches to that problem have been developed in this sense. The surveys [1, 3] are very complete compilations of the principal results obtained in this topic. Moreover, several Vizing-like results for some other kind of product graphs have been also published [5]. In E-mail address: ismael.gonzalez@uca.es (Ismael Gonzalez Yero) this article we introduce the notion of partial Cartesian product of graphs and obtain some corresponding Vizing-like results for the domination number and independence domination number of partial Cartesian product of graphs. We establish first the principal terminology and notation which we will use throughout the article. Hereafter G = ( V, E) denotes a finite simple graph. The complement of a set D is denoted by D. A set D is a dominating set if every vertex of D is adjacent to a vertex of D [4]. A dominating set D is called minimal if does not contain any dominating set as a proper subset. The domination number, y(G), is the minimum cardinality of any dominating set in G. We say that a set S is a Y(G)-set if it is a dominating set and |S| = y(G). The following result, due to Ore, can be found in [4]. Lemma 1.1. [4] A dominating set S is a minimal dominating set of a graph G = ( V, E ) if and only if for each u G S one of the following conditions holds: • u is an isolated vertex of S, • there exists v G V — S for which N (v) n S = {u}. 2 Results Let G and H be two graphs with set of vertices V1 = {ui,...,uni} and V2 = {vi,...,v„2}, respectively. We recall that the Cartesian product of G and H is the graph GDH = (V, E), where V = V1 x V2 and two vertices (ui, vj ) and (uk, v; ) are adjacent in GDH if and only if one of the following conditions is satisfied: • ui = uk and vj ~ v;, or • ui ~ uk and vj = v;. The strong product of G and H is the graph G H H = (V, E), where V = V1 x V2 and two vertices (ui, vj ) and (uk, v;) are adjacent in G H H if and only if one of the following conditions is satisfied: • ui = uk and vj ~ v;, or • ui ~ uk and vj = v;, or • ui ~ uk and vj ~ v;. For a vertex ui G V1, the subgraph of GDH or G H H induced by the set {(ui, v)|v G V2} is called an H-layer or an H-fiber. Similarly, for vj g V2, the subgraph induced by {(u, vj ) |u g V1} is a G-layer or a G-fiber. 2.1 Partial Cartesian product of graphs Let G and H be two graphs with set of vertices V1 = {u1,...,un2} and V2 = {v1,...,vni}, respectively. If S C V2, then the partial Cartesian product of G and H with respect to S is the graph GDSH = (V, E), where V = V1 x V2 and two vertices (ui, vj) and (uk, v;) are adjacent in GDSH if and only if one of the following conditions is satisfied: • ui = uk and vj ~ v;, or • ui ~ uk and vj = v; G S. Figure 1: The partial Cartesian product graph P4D{vi,v3}P4. An example of a partial Cartesian product graph is shown in Figure 1. Also notice that the partial Cartesian product of G and H is the spanning subgraph of GDH obtained by deleting the edges of GDH in the H-fibers that correspond to vertices from V2 \ S. Notice that if S = V2, then GDSH is the standard Cartesian product graph GDH. Also, note that if S is formed by only one vertex v, then GDSH is a particular case of the rooted product graph G Ov H defined in [2]. Domination related parameters of rooted product graphs were studied in [6]. Theorem 2.1. Let G be a graph of order n and let H be any graph. If S is a subset of vertices of H, then 7(GDsH) > n7(H) - n|S| + |S|y(G). Proof. Let Vi = [u,u2,...,un} and V2 be the vertex sets of G and H, respectively. Hence, we consider the set of vertices S C V2. Let D be a 7(GDSH)-set. For every Ui G Vi, let Di = [v G V2 : (щ, v) G D} and let Xi C Di be the set of vertices of H not dominated by Di. Hence, we have that Di U Xi is a dominating set in H and, as a consequence, 7(H) < |Di| + |Xi| for every is i G [1,..., n}. Hence, |D| = £ |Di| i=1 n > £(Y(H) -|Xi|) i=i n = П7(H) - £ |Xi|. i=i Thus, we have that n 7(GDsH) > nY(H) - £ |Xi|. (2.1) i=i Notice that if Xj = 0 for some j G [1, ...,n}, then there exists a vertex v G V2 such that (uj ,v) is dominated by a vertex (щ ,v) G D with l = j. Thus, v G S. Hence, for every vertex w G S let Dw = [ui G Vi : (ui,w) G ([ui} x Xi)}. Notice that Dw is a dominating set in G. So, y(G) < \Dw\ = n — |DwMoreover, we have that ЕП=1 \Xi\ = Ewes \Dw\. Now, by using (2.1) we have the following. n 7(GDsH) > nj(H) — E \Xi\ i=1 = n7(H) — E \Dw\ wes > n7(H) — E(n — Y(G)) n wes nY(H ) — n\S\ + \S\y(G). □ By taking a y(H)-set S in the above theorem we obtain a Vizing-like result for the domination number of the partial Cartesian product graph GDsH with respect to S. Corollary 2.2. Let G and H be any graphs. Then there exists a set S of vertices of H such that Y(GDsH) > y(g)y(H). To see that the above bound is tight we consider the following. We say that a graph G = ( V, E) satisfies property P if it has a Y(G)-set D such that D = X U Y, X n Y = 0, X dominates V — Y and Y dominates V — X. Let F be the family of all graphs satisfying property P. Proposition 2.3. Let G be a graph with domination number half its order and let H be a graph of the family F. Then there exists a set S of vertices of H such that Y (GDs H ) = y(g)y (H ). Proof. Since H = (V, E) is a graph of the family F, there exists a y(H)-set S such that S = X U Y, X n Y = 0, X dominates V — Y and Y dominates V — X. Also, as G has domination number n, where n is the order of G, we have that there exists a y(G)-set D, such that y(G) = \D\ = \D\ = n. Notice that the set (D x X) U (D x Y) is a dominating set in GDs H and, as a consequence, Y(GDsH) < \(D x X) U (D x Y)\ = \D\\X\ + \D\\Y\ = nY2H2 = y(G)y(H). The proof is finished by Corollary 2.2. □ Since for every set S of vertices of a graph H, the partial Cartesian product graph GDsH is a subgraph of the Cartesian product graph GDH, we have that y(GDsH) > Y(GDH). Hence, notice that if there exists a graph GDsH such that y(GDsH) = Y(G)y(H) and y(GDsH) > y(GDH), then the Vizing's conjecture is not true. In this sense, the above results lead to the following question. Question: Are there some graphs G and H such that some set S of vertices of H satisfies that y(GDsH) = y(G)y(H) and y(GDsH) > y(GDH)? Next we study the independence domination number of the partial Cartesian product of graphs. A set Y is an independent dominating set in a graph G, if Y is a dominating set and there no edge between any two vertices of Y. The minimum cardinality of an independent dominating set of G is the independence domination number of G and it is denoted by i(G). The following result from [6] will be useful to present our results. Lemma 2.4. [6] Let G = ( V, E ) be a graph. Then for every set of vertices A с V, i(G - A) > i(G) - |A|. Theorem 2.5. Let G be a graph of order n and let H be any graph. If S is a subset of vertices of H, then i(GDSH) > n(i(H) - |S|) + |S|i(G). Proof. Let V1 = {u1, u2,..., un} and V2 be the vertex sets of G and H, respectively, and let S с V2. Let Y be an i(GDSH)-set. For every u g Vb we define the following subsets of vertices of H: • Yj = {v G V2 : (uj,v) G Y}, • Si с S such that for every v g Sì, (uj, v) is independently dominated by some vertex (ui, v') g Y with v' g Yi, • R с S such that Ri = S n Yi. Thus, we have that for every i g {1,..., n}, Yi is an independent dominating set in (V2 - S + Rj + Sj). So, by Lemma 2.4 we have that |Yj| > i((V2 - S + Rj + Sj)) > i(H) -|S| + |Rj| + |Sj|. Therefore, we obtain that n n n |Y| = £ |Yj|> £(i(H)-|S| + |Rì| + |Sj|)= n(i(H)-|S|) + ^(|Rj| + |Sj|). (2.2) i=1 i=1 i=1 Now, for every vertex v g S we define the following subset of vertices of G: • Av = {uj g V1 : (uj, v) g Y}, • Bv = {ui g V1 : such that for every ui g Bv, (ui,v) g Si}, i.e., (ui,v) is independently dominated by some vertex (ui, v') where v' g Yi. Thus, we have that Av is an independent dominating set in (V1 - Bv ) and, by Lemma 2.4, we have that |Av| > i((V1 - Bv)) > i(G) - |Bv|. Moreover, notice that J]n=1(|Rj| + |Si|) = 2veS(|Av | + |Bv |). So, from Equation 2.2 we have the following. n |Y|> n(i(H) -|S|) + £(|Rj| + |Sj|) i=1 = n(i(H) - |S|) + ^(|Av| + |Bv|) v£S > n(i(H) - |S|) + ^(i(G) -|Bv| + |Bv|) v£S = n(i(H) - |S|) + |S|i(G). Therefore, the result follows. □ Notice that, if S is an i(H)-set, then the above result leads to item (i) of the next corollary . Moreover, if S is formed by a single vertex, then the above result leads to a lower bound for the independence domination number of rooted product graphs, which was also obtained in [6]. Corollary 2.6. Let G be a graph of order n and let H be any graph with vertex set V. Let s с v . • If S is an i(H)-set, then i(GDsH) > i(G)i(H). • If S = {v}, then i(GDs H ) = i (G ov H ) > n(i(H ) - 1) + i(G). 2.2 Partial strong product of graphs Now, if A с V1 and B с V2, then the partial strong product of G and H with respect to A and B is the graph G HA,B H = (V, E ), where V = Vi x V2 and two vertices (uj, v j ) and (uk, vi ) are adjacent in G HA,B H if and only if one of the following conditions is satisfied: • uj = uk and v j ~ vl, • щ ~ uk and v j = vl, • uj € A, v j € B, uj ~ uk and v j ~ vi, • uk € A, vi € B, uj ~ uk and v j ~ vi. Notice that if A = Vi and B = V2, then G HA,B H is the standard strong product graph G H H .A more restrictive condition on the partial strong product of graphs could be the following one. If A с Vi and B с V2, then the restricted partial strong product of G and H with respect to A and B is the graph G AHB H = (V, E), where V = Vi x V2 and two vertices (uj, v j ) and (uk, vl ) are adjacent in G A HB H if and only if one of the following conditions is satisfied: • uj = uk and v j ~ vl, • uj ~ uk and v j = vl, • uj € A, uk € A, v j € B, vi € B, uj ~ uk and v j ~ vi, • uj € A, uk € A, v j € B, vi € B, uj ~ uk and v j ~ vi. An example of a restricted partial strong product graph is shown in Figure 2. Lemma 2.7. Let G, H be any graphs and let A (respectively B) bea j(G)-set (respectively Y(H)-set). Then A x B is a minimal dominating set in G AHB H. Proof. The proof follows from Lemma 1.1 and the procedure of construction of the restricted partial strong product of graphs. □ We omit the proof of the following result, since it is straightforward. Theorem 2.8. Let G, H be any graphs. If A and B are dominating sets in G and H, respectively, then Y (G Ha,b H ) < y (G)y(H) and y(G a Hb H) < y (G)y(H ). v- • к О Vг О v, è и, и 2 ш и. Figure 2: The restricted partial strong product graph P4 {U1 ,U3} Kl{v2 ,v4} P4. It is straightforward to observe that the partial Cartesian product graph G\SH with respect to any set of vertices S of G is a subgraph of the restricted partial strong product graph G AH for any 7(G)-set A and any 7(H)-set B. Thus, according to results of Theorem 2.1, Corollary 2.2 and Theorem 2.8 there would exist a spanning subgraph F of the restricted partial strong product graph G A□ BH satisfying that 7(F) = 7(G)y(H). In this sense, the question is: Is always the Cartesian product graph G\H a spanning subgraph of the graph F ? If yes, then Vizing's conjecture is true. References [1] B. Bresar, P. Dorbec, W. Goddard, B. L. Hartnell, M. A. Henning, S. KlavZar and D. F. Rall, Vizing's conjecture: a survey and recent results, Journal of Graph Theory 69 (2012), 46-76. [2] C. D. Godsil and B. D. McKay, A new graph product and its spectrum, Bulletin of the Australasian Mathematical Society 18 (1) (1978), 21-28. [3] B. Hartnell and D. F. Rall, Domination in Cartesian products: Vizing's conjecture. In Domination in graphs, volume 209 of Monography and Textbooks in Pure and Applied Mathematics, pages 163-189, Marcel Dekker, New York, 1998. [4] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc. New York, 1998. [5] R. Hammack, W. Imrich and S. KlavZar, Handbook of product graphs, volume 66 of Discrete Mathematics and its Applications, CRC Press, 2011. [6] D. Kuziak, M. Lemarnka and I. G. Yero, Domination related parameters in rooted product graphs, Bulletin of the Malaysian Mathematical Sciences Society (2013), in press. [7] V. G. Vizing, The Cartesian product of graphs, Vycisl. Sistemy 9 (1963), 30-43. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 27-43 Cycle construction and geodesic cycles with application to the hypercube Paul C. Kainen Department of Mathematics and Statistics Georgetown University, Washington, DC 20057 USA Received 2 March 2013, accepted 23 April 2014, published online 9 June 2014 Abstract Construction of cycles in a graph is investigated, where cycles from particular subsets (such as bases) are added together so that each partial sum is also a cycle or each new cycle intersects the sum of the preceding terms in a nontrivial path. Starting with the geodesic cycles, a hierarchical construction is given. For the hypercube graph, geodesic cycles are characterized, and it is shown how hypercube geodesic cycles can be constructed in two steps from a special basis. Applications are given to inferring commutativity of a diagram in a groupoid from commutativity of some of its cycles. Keywords: Robust cycle basis, well-arranged sequence, geodesic cycle, Cayley graph, hypercube, forcing commutativity, groupoid diagram. Math. Subj. Class.: 05C38, 18B40 1 Introduction Let G be a graph. A cycle-subgraph of G is a connected, 2-regular subgraph. Let Cyc(G) denote the set of all cycle-subgraphs of G. Call S С Cyc(G) "weakly robust" if for every z G Cyc(G), there is an integer k > 1 and z1,... ,zk e S (not necessarily distinct) such that Z = zi + Z2 +-----+ Z k (1.1) and for 2 < j < k — 1 zi + z2 + ••• + zj e Cyc(G); (1.2) that is, S is weakly robust if it spans the cycle space of G in such a way that every cycle-subgraph has all partial sums remaining cycle-subgraphs (not just Eulerian graphs). Call S "robust" if it is weakly robust and if, in addition, for 2 < j < k, zj n (z1 + • • • + zj-_1) is anontrivialpath. (1.3) E-mail address: kainen@georgetown.edu (Paul C. Kainen) Of course, S = Cyc(G) has these properties vacuously, taking k = 1. Also, (1.3) (1.2). The problem of finding a cycle basis which is weakly robust seems to have first appeared in print in a paper of Dixon and Goodman [2]. Given a spanning tree in a connected graph, one can construct a basis of cycles using the non-tree edges. In [2], it was conjectured that such bases are weakly robust but this was disproved by Syslo [22]. Later, it was shown by Dogrusoz and Krishnamoorthy [3] that the set of region boundaries of a plane graph, excluding the unbounded region, constitute a weakly robust basis. The problem was rediscovered in [12], including the new question of finding a robust basis, where we showed that for complete graphs, the set of all 3-cycles containing a fixed vertex determines a robust basis. A basis which is not weakly robust (due to A. Vogt) was exhibited. We also found a basis for the bipartite complete graph (the set of all 4-cycles through a fixed edge), which was shown to be weakly robust by Ostermeier et al. [19]. Our motivation was category theory and graph theory, while the work of Stadler and co-authors [9, 11, 15, 16, 19, 8] is also motivated by applications, especially in biochemistry. These latter papers, as well as [12], use notation which is different from that used here. The properties of cycle bases and related sets discussed below were developed to permit the propagation of commutativity from a basis of cycles to all of the diagram; see [13], [14]. In this paper, it is shown that the commutativity of a d-dimensional hypercube diagram in a groupoid can be forced by the commutativity of a carefully chosen subset of the square faces, containing approximately a fraction of 4/d of the square faces. In contrast, [13] showed that commutativity cannot be blocked by any family with fewer than d - 2 squares in a d-cube; that is, if all but d - 2 squares in a d-cube diagram in a groupoid are known to commute, then the entire diagram is commutative. The paper is organized as follows: Section 2 gives some basic graph-theoretic, algebraic, and topological background. The next section reviews the basic types of well-arranged sequences of cycles, providing the machinery used in the remaining sections. Section 4 covers robustness and weak robustness, including the idea of iterating these constructions, and relates them to Cayley color graphs. In section 5 we give a brief proof that the basis of region boundaries in a plane graph is robust. Section 6 shows how geodesic cycles can build all cycles through an iterative process. Sections 7 and 8 apply only to the hypercube. Geodesic cycles in the d-cube are given a simple characterization and then they are constructed from a recursively defined basis. Section 9 derives the application to commutative diagrams in groupoids and shows that a square which is the sum of two commutative squares can be non-commutative if the two squares added do not intersect in a nontrivial path. Finally, in Section 10, other types of cycle bases are considered. 2 Basic concepts In this paper, graphs are finite, undirected, and simple (i.e., no loops or parallel edges). For basic graph theory, see, e.g., [7], [1]. A connected graph is Eulerian if all of its vertex-degrees are even. For G a graph, V(G) and E(G) denote the sets of vertices and edges, respectively, and we also write G = (V, E). Every graph has a representation as a family of points and closed straight-line segments (corresponding to the vertices and edges, resp.) embedded in 3-dimensional Euclidean space; each segment in the representation joins the two points corresponding to the edge which the segment represents. There is a unique topology which induces the usual topology of a closed interval on the straight-line segments, and this is called the underlying topology of the graph. A cycle is a graph which is 2-regular and connected. The underlying topology of a cycle is the unique closed connected 1-manifold, i.e., the circle. Let G be a graph and let Cyc(G) denote the set of all subgraphs of G which are cycles. A path is a graph which consists of a single vertex, or a connected graph which contains two vertices of degree 1 with every other vertex of degree 2. A nontrivial path is a path which contains more than one vertex. We write Pr for the path with r vertices. Let Z2 denote the field with 2 elements {0,1}; Z^ denotes the Z2-vector space of all functions from a set A to Z2; dim Z^ = |A|, the cardinality of A. Addition of functions is just coordinate-wise addition (modulo 2), so the zero element is the constant 0 function. To each function in ZA, one can associate the subset of A determined by those elements mapping to 1 and conversely for every subset of A, there is a unique element in ZA to which it is associated (the "characteristic function" of the set). The symmetric difference of two subsets of A corresponds to the vector space sum of the corresponding elements in ZA [1, p. 23]. If U is any subset of a vector space V, then span(U) is defined to be the intersection of all linear subspaces of V which contain U. Over the field Z2, the linear span of U is just the set of all sums of subsets of U ; if U is empty, its span is 0. If G = (V, E) is a graph, there is a linear map dG : zfG) ^ Z(G) given by dG(e) = v + w when e = vw G E (G). The members of Z.f(G) are called the 1-chains while members of Zf(G) are called 0-chains. The kernel (or "null space") Z (G) of the Z2-linear map dG is the subset of the 1-chains which map to the zero 0-chain. Thus, connected components of subgraphs of G induced by elements of Z (G) correspond to Eulerian subgraphs of G. By a theorem of Euler, H in Z (G) is an edge-disjoint union, hence sum, of cycles [1, p. 24], so Z (G) is the linear span of Cyc(G). So there is a basis for Z (G) consisting entirely of cycles, which is called a cycle basis. Addition in Z (G) is denoted +. One refers to the vector space Z (G) as the "cycle space" of the graph where the word "cycle" is used in the algebraic, rather than geometric, sense. The cyclomatic number b(G) of a graph G is the dimension of Z (G). It is well-known (and straightforward to show) that b(G) = q - p + k, where q,p, k denote the number of edges, vertices, and connected components of G. If G is connected, then for every spanning tree T of G, there is a basis B(G, T) which is the set of cycles ze formed by adding a single element e = vw in E(G) \ E(T). Indeed, as each pair v, w of distinct vertices in G are joined by a unique path in T, T + e has a unique cycle. Each of these cycles has an edge contained by no other cycle, so the set of cycles must be independent. But the cardinality of this independent set is maximal, for T contains p - 1 edges, so |B(G, T) | = q - p + 1, and so it is a basis, the spanning tree basis, 3 Well-arranged sequences of cycles If z, z' G Cyc(G), we write z||z' and call z, z' compatible if z n z' = P, where P is a nontrivial path. This relation on the set of cycles is symmetric and irreflexive. For two compatible cycles z, z', the underlying topology of z + z' is the connected sum of the underlying topologies of the components, so z + z' must be a cycle as well. We give a direct argument avoiding the reference to connected sum after introducing a useful notation. If z is any cycle and P is a nontrivial path contained in z, then there is a unique nontrivial path P' also contained in z such that z = P U P' and P n P' = 2K1 (i.e., P and P' intersect in two isolated vertices). We write z - P for P'. Lemma 3.1. The sum of two compatible cycles is a cycle. Proof. Let z, z' be two cycles with z n z' = P a nontrivial path. Then z + z' = (z - P) U (z' - P). The paths z - P and z' - P intersect in the two endpoints of P, so their union is a cycle. □ Let z G Cyc(G); suppose S С Cyc(G) and span(S ) = Z (G); e.g., S could be a cycle basis. A z-sequence in S is a sequence a = (s1, s2,..., sk) G Sk such that k z = s 1 + S2 +-----+ Sk = £ Si. i= 1 We do not insist that the entries in a need all be distinct. Of course, for every cycle z there are z-sequences in S with all members distinct since S spans every cycle, but in order to achieve further constraints on the sequence, it may be necessary to allow repetitions of members of the sequence. When there are no repetitions, it will be indicated. For a sequence a = (s1, s2,..., sk) g Sk, there are several notions of "well-arranged": a is well-arranged with respect to intersection (wai) if for each j, 2 < j < k, s j intersects the previous partial sum si in a nontrivial path, that is, s j || s^ for 2 < j < k; a is well-arranged with respect to topology (wat) if each partial sum is a cycle; that is, if the partial sums are topologically constant. These are the only variants of well-arrangedness which we shall need below but for completeness we mention two more. a is well-arranged with respect to connectedness (wac) if each partial sum is connected; a is well-arranged with respect to degree (wad) if each partial sum is regular of degree 2. By Lemma 3.1, wai wat. However, the converse is false. Noncompatible cycles may sum to a cycle (see, e.g., the example at the end of section 6), so there can be z-sequences which are well-arranged with respect to topology but not with respect to intersection. Note that wat wac and wat wad trivially. Conversely, if wad holds for the z-sequence a, then a satisfies a weaker intersection condition that for each j, 2 < j < k, s j intersects the partial sum j=i si in a subgraph with no isolated vertices. Further, this weak intersection condition, in turn, implies wad. 4 Robustness and weak robustness Let G be a graph and let S С Cyc(G). We define the robust closure p(S) (or weak robust closure pw (S)) of S with respect to G to be the set of all cycles in G which are the sum of a wai (resp., wat) sequence of elements in S. A set S of cycles is called robust (weakly robust) if p(S) = Cyc(G) (resp. pw (S) = Cyc(G)). Of course, a weakly robust (and so also a robust) set of cycles is necessarily spanning: span(S ) = Z (G). When S is a basis for Z (G), then every cycle can be written as the sum of a unique subset of the basis, with no element repeated. This non-repetition property of the sequence is not required for the definition of robust and weak robust closure. However, no examples are known where every wai (or wat) sequence which sums to some cycle z has repeated terms. Hierarchical iteration is achieved by taking the robust span of the set of cycles produced by the previous stage. Let p0(S) = S and, for integer a > 1, put pa(S) := p(pa-1(S)). Then pa (S) C pb(S) when a < b, for a, b nonnegative integers. Also, if S C T, then p(S) C p(T). We note the following for use in Section 9. Let T C Cyc(G) for some graph G. Call T cooperative if (z, z' e T and z||z') z + z' e T. According to Lemma 3.1, the set of all cycles is cooperative. From the definition of iterated robust closure and our remarks there, if G is any graph, one has the following. Lemma 4.1. If S CT C Cyc(G), T is cooperative, and к is a nonnegative integer, then pk(S) CT. Hierarchically iterated robust closure and the lemma above are applied in Cor. 9.3 to obtain results on the propagation of commutativity in general groupoid diagrams, especially when the diagram has the scheme of a hypercube. (Definitions are given there.) These results can be expressed in terms of Cayley color graphs [24]; see [16], [19], where the concept is used but not the name. Recall that the Cayley color graph of a group A with respect to a subset SCA is the digraph Г(А, S) with A as its vertex-set, where for vertices v, w, there is an arc a = (v, w) if and only if there exists an element u e S such that w = v + u. (We write "+" as we are only considering the Abelian groups underlying vector spaces.) The arc a = (v,w) is colored by assigning to it the unique u satisfying the equation w = v + u, and so the Cayley color graph is the ordered pair consisting of the digraph and the corresponding arc colors. Let A be the abelian group determined by Z (G). Each element has order 2, so the Cayley color graph is symmetric and can be replaced by an undirected graph. To avoid loops, one assumes that the identity element of the group is not in the set S. Plainly, S is spanning if and only if r(A, S) is connected; see, e.g., [19, Lemma 3.3]. Let G be a graph, let S C Cyc(G) with span(S ) = Z (G). Let Г Cyc(G, S ) denote the subgraph of r(A, S) induced by Cyc(G) C A. Given two cycles z, z' in G there is a z'-z-path in rCyc(G, S) joining them if and only if there is a wat z-sequence a in S and such that z' = si. It is convenient to take z' to be the null-set of edges which is 0 in the vector space. The set of all cycles z reachable from 0 by such paths is exactly the weak robust span of S. The graph rCyc(A, S) is connected if and only if there is a rCyc(A, S)-path joining 0 to each z e Cyc(G), that is, if and only if S is weakly robust. The robust span of S is the set of all vertices in the connected component of 0 in the edge-induced subgraph r'Cyc(G, S) of rCyc(G, S) determined by those edges zz' where z||z'. In this case, the partial sums are successively modified by replacing a single subpath P with a complementary path P' = z — P, where P C z, for z some member of S. Instead of robustness or weak robustness, given any family S of cycles which spans Z (G), one might investigate the least number of connected components, the least maximum degree, or the least number of distinct topologies which occur along the rCyc(A, S) (or Г C (A, S )) paths from 0 to z, maximized or totaled over all cycles z in G. One may also focus only on some cycles which need to be built up. See [15] for an application to random networks and also Proposition 10.1 below. 5 Planarity and robustness For plane graphs, we find a robust basis; cf. [3]. In the following, we do not distinguish between subgraphs of a plane graph and the corresponding subsets of the plane. Let G be a 2-connected plane graph. To prevent pathological embeddings, assume that each edge of G is embedded as a piecewise-linear curve (with a finite number of segments). Then R2 \ G is a disjoint union of open disks and, using the 2-connectedness of the graph, the closure of each open disk is a closed disk and the boundary of each closed disk is a cycle. Each plane graph contains exactly one unbounded region, containing points which are arbitrarily far from any point of the graph; all the other regions are called bounded. See, e.g., [1, Chap. 4]. Let R(G) denote the set of all bounded regions. Then B(G) := {dr : r G R(G)} is a basis of the cycle space of G . It is spanning since if z g Cyc(G), then z = J2 dr, summing over all regions r contained within z. By Euler's Formula, the number of all regions is equal to q — p + 2, so |B(G) | = b(G) and hence B(G) is a cycle basis. The interior int(P) of a nontrivial path P is the path minus its two endpoints. A topological path is any curve which is homeomorphic to the closed unit interval [0,1]. A subset of the plane is topologically path-connected if any two of its points can be joined by a topological path which is the union of a finite number of straight line segments. Lemma 5.1. For any 2-connected plane graph G and z G Cyc(G), either z = dr for some r G R(G) or there exists r G R(G) such that z n dr is a nontrivial path. Proof. Assume z = dr for all r G R(G). Then dr n z is a disjoint union of paths, some of which can consist of an isolated vertex. Let R1, R2 denote the sets of regions r in R(G) which are contained within z and for which H = dr n z has at most 1, resp. at least 2, connected components. If R2 is empty, then for each edge e in z, the unique region r whose intersection with z includes e must satisfy dr n z is a nontrivial path. Suppose R2 is nonempty. If P is any path subgraph of z joining two successive components of z n dr, then r separates the interior of P from the interior of z - P in the sense that any topological path in the plane joining apointin int(P ) to a point in int(z —P ) must intersect dr. See [1, Lemma 4.1.2] for a formal proof. Let P * be a path subgraph of z joining two successive components and of minimum possible length. Then for any edge e in P*, there exists a unique region r' for which e G E (z n dr'). If z n dr' is not contained in P *, then the path-connected set r' would allow a topological path between some point in the interior of P * and some point in the interior of z — P *. But this is impossible by our remark about the separating property of r. Hence, z n dr' C P * .By minimality of P *, r' must intersect z in a single component. Since the intersection contains an edge, it is a nontrivial path. □ Theorem 5.2. The basis R(G) of a 2-connected plane graph G is a robust basis; in fact, each cycle is the sum of a wai sequence with no repeated elements. Proof. Let t be the number of regions contained in z; we prove the result by induction on t. For the basis case, t = 1 and the wai sequence has the single term z. If t > 1, then by Lemma 5.1, there is a region r in R(G) contained in z such that dr n z is a nontrivial path. Hence, zZ = z + dr e Cyc(G) and z' contains every region contained in z except for r. By the induction hypothesis, there is an ordering of the regions contained in z' so that z' = dri + dr 2 +-----+ drt-1 and the sequence (dri,..., drt—i) is well-arranged w.r.t. intersection. Therefore, concatenating one more term, dr, gives the desired wai sequence of members of R(G). □ 6 Geodesic cycles can build all cycles If z is any cycle (or path), let l(z) := length(z) denote the number of edges. The girth g = g (G) of a graph G is the length of a shortest cycle [1, p. 8]. A subgraph H C G is geodesic (or isometric) if for all v, w vertices of H, distH (v, w) = disto (v, w), where for a,b e V (H ), distH (a, b) is the distance from vertex a to vertex b in H - i.e., the length of a shortest a-b-path in H (and similarly for G). In K4, any 3-cycle is geodesic but any 4-cycle is not geodesic. It is clear that a cycle of minimum length is necessarily geodesic, so geodesic cycles always exist in any non-forest. The length of a geodesic cycle can't exceed twice the diameter by more than 1 as is shown in [1, proof of Prop. 1.3.2]. Hence, in a bipartite graph, every geodesic cycle has length at most twice the diameter. Let Geo(G) denote the set of geodesic cycles. We now show that for all graphs G, every cycle belongs to an iterated robust span of the geodesic cycles and bound the number of iterations needed; cf. [4]. Theorem 6.1. For every graph G and every z e Cyc(G), one has z e pe(z)—9(o)(Geo(G)) (6.1) Proof. Suppose G is a fixed arbitrary graph and z is any cycle of G. We write k = k(G, z) = l(z) — g(G). The statement of the theorem holds when k = 0 since then the cycle must be of minimum length, hence geodesic, and so in p0(Geo(G)). Suppose the theorem holds when k(G,z) < n, where n is some positive integer and let z be a cycle of G with length n + g(G) (i.e., so that k(G, z) = n). If z is not a geodesic cycle, there exist non-adjacent vertices v, w in z and a v-w-path P in G, intersecting z in exactly the two vertices v, w (and no edges) such that the length of P is less than the distance between v and w in z. Let P1,P2 be the two disjoint v-w-paths in z and put zi = P U Pi, i = 1, 2. Since (z1, z2) is a wai sequence with sum z, z e p({z1 ,z2}). But both z1 and z2 are strictly shorter than z so by the inductive hypothesis, z1,z2 e pk (Geo(G)), where k' = max{£(z1), £(z2)} — g(G) < k — 1. Therefore, we have z in p1+k (Geo(G)) C pk(Geo(G)) which completes the induction. □ Let c(G) denote the circumference of G, which is the length of a longest cycle. Corollary 6.2. For every graph G, Cyc(G) C p 2, such that Ф(ш) = a * a, (7.1) where a is a string which is a permutation of the elements of S. Proof. Let z be any geodesic cycle in Qd. Then for any two diametrically opposite vertices v, w of z, the two distinct v-w-paths P, P' contained in z must both be geodesic paths. Let t be the closed walk P' followed by Pop. By the remark above, both P and P' correspond to permutations of sets S, S', resp. But S = S' as the walk is closed. Hence, any r in [d] appears either twice or not at all in the string Ф(т). If the two appearances of some r G [d] were not at diametrically opposite edges e, e' of z, then there exist diametrically opposite vertices v', w' G V (z) such that both e and e' are contained in one of the two geodesic paths in z which is impossible. Conversely, for a walk of length 2j satisfying the permutation condition, any subpath of length < j corresponds to a permutation and is geodesic so the cycle is also geodesic. □ 8 Building geodesic cycles in the hypercube Since Qd is bipartite, shortest cycles have length > 4. For distinct i,j in [d], the closed walk ш = (v, v + ui,v + ui + u j, v + u j, v) is a 4-cycle and Ф(ш) = (i, j, i, j). Moreover, any 4-cycle, being geodesic, must be of this form. Call 4-cycles squares and let Sq(Qd) be the set of all squares in Qd. For convenience, during this section we shall write ij for any square s such that Ф(в) = (i, j, i, j), for distinct i, j G [d]. We will use the notion of a strip of squares in some graph G. This is a sequence a = (si,s2,...,sk ), (8.1) where each sj is a distinct 4-cycle subgraph of G, no two squares intersect unless one is the successor of the other, and for 2 < j < k — 1, sj intersects its predecessor sj_i and its successor sj+1 in disjoint, that is, opposite, edges. The length of a strip of squares is the number of squares in the sequence. If the length of a strip of squares a as in (8.1) is k > 2, then we call the strip nontrivial. For a nontrivial strip there are distinguished start and stop edges for a: the start edge is the edge of s1 opposite to the edge s1 П s2, and the stop edge is the edge of sk opposite to the edge sk-1 n sk. The ties of a nontrivial strip of squares is the set consisting of the start and stop edges, together with all intersection edges sj n sj+1, 1 < j < k — 1. Let G(a) := U k=1 s j denote the graph determined by the strip of squares. A strip of length 1 has graph which is a 4-cycle and so is isomorphic to Q1 x P2. The following results can be easily proved by induction on k. Lemma 8.1. Let a be a strip of squares of length k > 2. Then (i) G(a) is isomorphic to Q1 x Pfc+b (ii) the ties of a correspond to the edges Q1 x v for v G V (Pk+1), and (iii) the squares s1,..., sk are a robust basis of G(a). Lemma 8.2. In a hypercube, all ties of a strip of squares have the same active coordinate. In the following, assume that d > 3, else everything is trivial. For 1 < j < d, define strips aj in Qd as follows. Let s2 be the unique 12 square which contains 0, put aj = s2, and let [u2,u2 + u1] be the "stop edge" of s2. For 2 < j < d, having defined aj-1 let j j—1 aj := aj * sj, where s j is the unique 1j square through the vertex u2 + • • • + uj-1. Let zj be the sum of the squares in aj. that is, zj := S2 +-----+ s j. We prove by induction on j that Ф (zj ) = (1,...,j, 1,j,j - 1,..., 2). (8.2) For j = 2 the result holds and for 2 < j < d - 1, adding sj+1 to z j replaces the stop edge of aj with a sequence of three edges active consecutively in coordinates j + 1,1, j + 1. Thus, Ф^1) = Ф^' + s j+1) = (1,...,j + 1,1,j + 1,j,j - 1,..., 2), establishing the induction. Let a1 = ad and put z1 := zd. Then Ф^) = (1, 2,..., d, 1, d, d - 1,..., 3, 2). Thus, z1 is the union of two paths P, P1 from 0 to I, where Ф(Р) = (1, 2,..., d) and Ф(Р1) = (2,..., d, 1). Hence, z1 is a cycle of length 2d, but it is not geodesic since the intersection edges of a1 join non-adjacent vertices of z1. We now show that every geodesic cycle in Qd is in the robust span of the set of squares by continuing the above construction. Theorem 8.3. Let z be any geodesic cycle of length 2j in Qd, d > 3. Then there exists a wai z-sequence of distinct squares z1, z2,..., zm, where m = j (j - 1)/2. So z G p(Sq(Qd)). Proof. By the homogeneity of Qd, it suffices to show that, for each positive integer d > 3, the unique geodesic cycle z in Qd through 1 with Ф^) = (1,2,..., d, 1, 2,..., d) is the sum of a sequence a of squares which is well-arranged w.r.t. intersection. For 1 < j < d - 1, let a j denote a sequence of squares of length d - j of the form aj := (j j + 1 j j + 2,..., j d); put a := a1 * a2 * • • • * ad-1, the concatenation. Each aj is a strip of squares, the strips are disjoint in the sense that no square belongs to more than one, and every square does belong to a strip. We will show that this sequence of squares is wai and has sum z. In the sequel, as we add squares to z1 = P U Рь the path P will be unchanged while the path P1 will be successively transformed, but without changing its length. The sequence a2 = (s23, s24,..., s2d) is defined as follows. Let s23 be the unique 23 square which includes the vertex note that s23 intersects P1 in the path P1,23 = (0,u2,u2 + u3) which are the first two edges of P\. Adding s23 to z1 produces a new cycle z1j23 which is the union of P and the path P1j23 obtained by replacing (0, u2, u2 + u3) by (0, u3, u2 + u3) in P1. Now there is a unique 24 square s24 which includes the vertex u3, and s24 intersects P1,2 3 in the path P1,24 = (u3, u2 + u3, u2 + u3 + u4) which constitutes the second and third edges of P1j23. Put z1,24 = z1j23 + s24. Then z1,24 = P U P1j24, where 4 5 d d P1,24 = (0, u3,u3 + u4, ^uj, ^uj,..., Eu, Eu). j=2 j=2 j=2 j=1 Now there is a unique 25 square through the vertex u3 + u4 intersecting P1j24 in its third and fourth edges, and so forth up to 2d. Adding the sum z2 of the squares in a2 to z1 produces Ф^2) = Ф^ + z2 ) = (1, 2 ,...,d, 1, 2, d, d - 1,..., 3). Proceeding in this fashion, z = zd, Ф(z) = (1,..., d, 1,..., d), and a is a wai z-sequence as each square after those in the initial subsequence a1 intersects the sum of the preceding squares in a path of length 2. □ Thus, the set Sq(Qd) of all squares in Qd robustly spans the set Geo(Qd) of all geodesic cycles in Qd, Geo(Qd) C p(Sq(Qd)). But Sq(Qd) contains d(d - 1)2d-3 squares, which is asymptotically d/4 times the size of a cycle basis. Indeed, b(Qd) = 1 + d2d-1 - 2d = 1 + (d - 2)2d-1; e.g., [7, pp 22-23, 38-39]. In the graph Q1 x G, as V(Q1) = {0,1}, G is isomorphic to 0 x G under the mapping v ^ (0, v), and we write 0 x H for the image under this isomorphism of a subgraph H of G in 0 x G and similarly for 1 x H. Also, if e G E (G), we write e for the corresponding subgraph (which is isomorphic to Q1), and for e G E(G), each subgraph Q1 x e in Q1 x G is a 4-cycle. We shall use the decomposition E(Qd) = E(Qd) U E (Qd) и Ed, (8.3) where Q0 = 0 x Qd-1 and Qd = 1 x Qd-1 denote the "bottom" and "top" subgraphs of the d-cube, Ed := E(Qd) \ (E(Q0) U E (Qd)) denotes the "side" edges of the d-cube. Similarly, there is a decomposition of the squares Sq(Qd) = S0 U S U Sqd, (8.4) where Sj := Sq(Qjd) for j = 0,1 and Sq'd = Sq(Qd) \ (S0 U Si) (the set of all "side" squares of Qd). Each element s in Sq'd is the product of Q1 with the unique edge e in E(Qd-1) corresponding to the intersection of s with Qd. Now we can define the Kainen basis of Qd [12]. Let BK (Qd) be the following recursively given collection of squares in Qd. For d = 0 and d = 1, the set is empty, and Bk(Q2) = {Q2}. Having defined Bk(Qd_1) for d > 3, put Bk (Qd) := Bk (Qd) U Sq'd, (8.5) where BK(Qd) is the cycle basis of Qd corresponding to BK(Qd_1) under the natural isomorphism. For example, BK (Q4) consists of five squares in one of the 3-cube faces and all twelve side-squares. Lemma 8.4. The set BK (Qd) is a cycle basis for Qd for d > 2. Proof. By induction on d; trivial for d = 2. Now let d > 3. Then BK (Qd) is independent since by induction the squares in BK(Qd) in the decomposition (8.5) are independent, while the squares in the sides, Sq'd, are independent of those in the bottom and also of each other as each contains a unique edge in Qd-1. But BK (Qd) has the cardinality of a basis (and so is a basis). Indeed, using (8.5), we have |Bk (Qd)| = |Bk (Qd_1))| + |E (Qd_1)| = 1 + (d—3)2d_2 + (d—1)2d_2 = 1+(d—2)2d-1, where the second equality again uses the inductive assumption that BK(Qd_1) is a basis. □ We now show that this basis is sufficient to robustly span every square in Qd. Theorem 8.5. For d > 2, Sq(Qd) С p(Bk(Qd)). Proof. As before, the basis case d = 2 is trivial. Every square in Qd which is on the bottom is in p(BK(Qd_1)) by the induction hypothesis and p(BK(Qd_1)) С p(BK(Qd)) and the side-squares are in BK (Qd) by definition. Thus, we only need to take care of the squares s on the top; that is, s e S1 = Sq(Qd). Each such square is contained in a unique subgraph Q3 of Qd such that s = Q1,. Let s' = Q3. Then s' e Sq(Qd) so by induction there exists a' a wai s'-sequence in BK (Qd_1) С BK (Qd), and we may append to this sequence the 4 side-squares of Q3 (in any order) to produce a wai sequence in BK(Qd) which sums to s. □ Using Theorem 6.1, we see that for hypercubes every cycle is in the iterated robust closure of BK (Qd). Corollary 8.6. For d > 2 and z any cycle in Qd, z e p£(z)_2(BK(Qd)). 9 Application to commutativity in groupoids The original motivation for building up cycles using wai sequences was to show that the commutativity of certain diagrams in a groupoid category can be inferred from the commutativity of a small minority of the faces of the diagram [13]. After defining these terms, we use the results of the previous sections of this paper to prove the above claim and we show that a similar inference cannot be made using only wat sequences. For a full discussion of the concept of a category, see, e.g., Mac Lane [17]. Briefly, a (small) category C consists of a set Obj(C) of objects, a set Mor(C) of morphisms, and two functions ("domain" and "co-domain") dom, cod : Mor(C) ^ Obj(C). (A morphism thus models the notion of a structure-preserving function between two mathematical objects.) An ordered pair (a, b) of morphisms is called composable if dom(b) = cod(a). In addition, C is required to have a law of composition which is a function from the set of composable morphism pairs to the set of morphisms denoted by (a, b) ^ ab, where dom(ab) = dom(a) and cod(ab) = cod(b). Often the notation for composition is in reverse order, emulating composition of functions. It is convenient to write C(x, y) for the set of all morphisms in C with domain x and co-domain y, and a : x ^ y is another notation for a G C(x,y) Typically, the set of morphisms between a given pair of objects has many elements but this set can also be empty or a singleton. For any triple x, y, z of objects of C the law of composition gives a function C(x, y) X C(y, z) ^ C(x, z). Composition in a category C must be associative: if (a, b) and (b, c) are composable pairs of morphisms in C, then (ab, c) and (a, bc) are both composable pairs and we require (ab)c = a(bc). It follows by induction that one can define a unique composition a\a2 • • • ak for any finite sequence (a1,a2,... ,ak ) of morphisms for which each successive pair (ai, ai+1) is composable. The final condition in the definition of a category C is that for each object x, there is an identity morphism 1x G C(x, x) such that if x, y are any two objects in C, then for any morphism a in C(x, y), aly = a = 1xa. Typical examples of a category are a set of topological spaces and continuous maps, or a set of vector spaces and linear maps, etc., provided that the composition of any two morphisms remains in the set, and of course that the identity morphisms are included. A directed multigraph D = (V, A, Ф) is a set V of vertices, a set A of arcs, and a mapping Ф : A ^ V x V which associates to each arc an ordered pair (v, w), where v is its source and w its target. Arc-pairs (a, a') are composable if the target of a is the source of a' and an arc sequence is a dipath if each successive pair is composable. A v-w-dipath is a dipath where the first arc has source v and the last arc has target w. A diagram S in a category C is a directed multigraph D = (V, A, Ф) (called the scheme) and functions fv, fA such that fv : V ^ obj(C), fA : A ^ mor(C) such that if ^(a) = (v,w) and fv(v) = x,fv(w) = y, then fA(a) G C(x,y). Unless necessary, we shall not distinguish between vertices of the diagram and objects of the category, nor between arcs and morphisms. A dipath in the diagram corresponds to a composable sequence of morphisms in the category. A face of the diagram is a distinct pair of v-w-dipaths, for some objects v, w, and this face is commutative if the two paths give rise to the same v-w-morphism, using the law of composition in the category. The face is said to "commute." The diagram is commutative if all of its faces commute. The importance of commutative diagrams is that they can be used to define certain algebraic properties, such as co-associativity, and a basic theory of algebraic syntax can be built up from commutative diagrams - see, e.g., [18] and [5]. For an explicit recent example from computer science, see [21, p. 13, diagram for Remark 1.2.6]. A groupoid is a category G in which every morphism is invertible; i.e., for every mor-phism a, there is a unique morphism a-1 such that composition in either order gives 1. Groupoids are generalizations of ordinary groups and they have become of some interest in a number of fields recently, spreading far from initial applications in topology; see, e.g., [23]. A diagram in a groupoid G can be extended by including for every arc a in the diagram a reverse arc aop and if the morphism a is assigned to a then in the extended diagram, the inverse morphism a-1 is assigned to aop. If the original diagram commutes, then so does the extended diagram and the reverse implication is trivial. Hence, we shall assume that all diagrams in a groupoid category are symmetric. Any undirected cycle in a groupoid diagram can be parsed into a closed walk by choosing one of the vertices and then proceeding either clockwise or counterclockwise. The cycle is said to g-commute if the composition along the walk is the identity; this does not depend on how the cycle is parsed; e.g., if abc = 1, then 1 = a-11a = bca, etc. A diagram in a groupoid category will be called g-commutative if and only if all of its cycles g-commute. Lemma 9.1. If a groupoid diagram is g-commutative, then it is commutative. Proof. Suppose we have a g-commutative diagram. Let P, Q be two directed paths from v to w in the diagram and let a, ß be the morphisms induced by P and Q, resp. It suffices to assume P and Q intersect only in v and w. Let z be the cycle formed by the union of P and Q, parsed to begin at v, follow P, and then Q (in reverse order). Then z induces aß-1. So the morphism induced by z is 1 if and only if a = ß. Thus, g-commutativity implies ordinary commutativity. □ The converse is false. For example, one can have four objects w, x, y, z with morphisms f : w ^ x, g : y ^ x, h : y ^ z, i : w ^ z. The corresponding diagram is automatically commutative because it has no faces but if it is in a groupoid it is g-commutative only if fg-1hi-1 = 1, i.e., fg-1 = ih-1. Proposition 9.2. The g-commutative cycles in a groupoid diagram form a cooperative set. Proof. Let z, z' be commutative cycles in a groupoid diagram, with P = z n z' a nontrivial path from v to w. Let a be composition along P, ß along z — P, and ß' along z' — P. Then aß = 1v and aß ' = 1v But also ß-1a-1 = 1v. Hence, 1v = ß-1a-1aß' = ß-1ß'. Thus, z + z', which is the cycle (z — P) U (z' — P) is commutative. □ Using Lemma 4.1 and Proposition 9.2, by Theorem 6.1 and Corollary 8.6, we get: Corollary 9.3. A diagram in a groupoid is g-commutative if its geodesic cycles g-commute. A Qd-diagram in a groupoid is g-commutative if each z e BK (Qd) g-commutes. The sum of two commutative cycles can be a noncommutative cycle when the cycles intersect in more than one nontrivial path. Label the vertices of a complete graph on 4 vertices A, B, C, D in clockwise order, where all vertices correspond to a single object X in the groupoid, and take all morphisms to be 1X, except for the two diagonal arcs from C to A and from D to B, respectively, which represent some morphism x : X ^ X, where x2 = 1X. (This is a mild condition on the groupoid, satisfied for instance by the symmetric group Sn for n > 3.) Consider the two squares A, B, C, D and A, B, D, C ; going around the first square gives a 4-fold composition of identities, while the second square gives x О 1x О x-1 О 1x = 1x . The two squares intersect in edges AB and CD and the sum of the two squares is the square A, D, B, C which induces the morphism x2 = 1X. Thus, propagation of commutativity to the sum can fail when cycles intersect in two disjoint paths. 10 Remarks This section is essentially an appendix which compares other types of cycle basis to the one we have constructed. We thank one of the referees for suggesting some of the references given here and for other helpful comments. A general method for generating cycle bases for Cartesian product graphs, was described by Hammack [6]. It constructs a cycle basis for G x G' for each pair of graphs G, G' with given spanning trees T, T' and cycle bases B, B' for G, G', resp. Then there is a Hammack basis BH(G, B, T, G', B', T') := F U G U G', where F := {e x e' : e e E(T), e' e E(T')}, G := {z x y : z eB,y e V (G')}, G' := {x x z' : x e V(G),z' e B'}. For the hypercube Qd, it is reasonable to take G = Qd-1 and G' = Q1. So T' = Q1 but T could be an arbitrary spanning tree of Qd-1. There is a natural recursive choice for T a spanning tree of Qd-1 when d > 3: just keep adding in all of the "side" edges. For example, the spanning tree constructed for Q3 looks like a "U" standing on 4 legs. If d > 4, the Hammack bases (allowing all possible spanning trees) do not include the Kainen basis since the latter uses only bottom squares, but all (d - 1)2d-2 side-squares, while the former type use both top and bottom squares but only 2d-1 - 1 side squares. For example, BH (Q4) will have 5 squares in both Q4 and Q4 but only 7 side-squares (cf. supra Lemma 8.4). Other iterated Cartesian product graphs have been explicitly studied, e.g., in [9]. Minimum cycle bases (in the sense of having minimum total sum of the lengths of their members) for Cartesian product graphs were constructed by Imrich and Stadler [11]. Like the Hammack bases, these depend on a spanning tree and a cycle basis, and also a vertex, for each factor of a Cartesian product. The Imrich-Stadler basis for G x G' has the form Bis (G, B, T, x, G', B', T ', y) := Bn U Gy U G'x, where B, T, x (B', T', y) are cycle basis, spanning tree, and vertex for G (and for G', resp.), and Bu := {e X e' : e G E (T ), e' G E (G')} U {e x e' : e G E (G), e' G E (T ')}, := {z X y : z G B}, GX := {x X z' : z' G B'}. If G' = T', then Bu = {e x e' : e G E(G),e' G E (T')} and G'x is empty. So BIS(G X T') doesn't depend on T or x. When G = Qd-1 and T = Qb this is just BK (Qd ) so BIS is a generalization of BK. When the cycle bases of G and G' are minimal and both G and G' have girth at least 4, [11, Thm. 8] shows that BIS(G x G') is minimal as well. When G' is a tree, we can get a robustness-related result. In the graph G X T, T a tree, a cycle z contained in a subgraph G x v' for some v' G V (T) is called a level cycle. Proposition 10.1. Let G be a graph with spanning tree T, cycle-basis B, and v0 G V (G). If B is robust, then every level-cycle in T x G is in the robust span of the basis B' B' := {v0 x z : z G B} U {e' x e : e' G E (T ), e G E (G)}. Proof. Suppose that T has t vertices and G has p vertices and q edges. Then it is easy to calculate that b(T x G) = qt - p + 1 = |B'| and B' is a basis by [11, Thm. 4]. We prove the robustness claim by induction on the number t of vertices of T. When t = 1, B' = B so the result holds. Let T be any tree with t vertices, t > 1. Then there is a vertex v in T such that v has degree 1. Let T' = T - v. Now consider any level cycle z' in T x G. If z' is in w x G for some w G V (T '), then by the induction hypothesis, there is a z'-sequence from B' which is wai. Otherwise, if z' is contained in v x G, then z' = v x z for some z G Cyc(G). Let v' be the unique vertex of T which is adjacent to v. For the cycle z'' = v' x z, there is a z''-sequence a' from B' which is wai. If the edges in z are followed in a closed walk, say e1,..., ek and if e = vv', then a = a' * (e x ei, .. ., e x e^) is a wai sequence from B' which sums to z'. □ Recently, convex cycle bases were studied in [8]. A subgraph H of G is convex if for every minimal-length G-path P between vertices v,w G V (H ), P is a subgraph of H. In a hypercube, squares are convex subgraphs. The Imrich-Stadler basis for G x H is convex when the bases for G and H are convex. They note that the only convex cycles in a hypercube are the squares. Geodesic cycles in the hypercubes are a special case of planar partial cube and the latter have been characterized in [20]. References [1] R. Diestel, Graph Theory, Springer, Berlin, 2005. [2] E. T. Dixon and S. E. Goodman, An algorithm for the longest cycle problem, Networks 6(2) (1976), 139-149. [3] U. Dogrusoz and N. Krishnamoorthy, Enumerating all cycles of a planar graph, Parallel Algorithms and Applications 10 (1996), 21-36. [4] A. Georgakopoulos and P. Sprüssel, Geodetic topological cycles in locally finite graphs, The Electronic Journal of Combinatorics 16 (2009), #R144 (18 pages). [5] J. A. Goguen, A categorical manifesto, in Mathematical Structures in Computer Science 1 (1991), 49-67; http://www.cs.ox.ac.uk/files/3395/PRG72.pdf [6] R. Hammack, Cyclicity of graphs, Journal of Graph Theory 32(2) (1999), 160-170. [7] F. Harary, F., Graph Theory, Addison-Wesley, Reading, MA, 1970. [8] M. Hellmuth, J. Leydold and P. F. Stadler, Convex cycle bases, Ars Math. Contem. 7(2014), 123-140. [9] W. Imrich and S. KlavZar, Product Graphs: Structure and Recognition, Wiley, New York, 2000. [10] W. Imrich, S. KlavZar, and D. F. Rall, Graphs and Their Cartesian Product, A. K. Peters, Wellesley, MA, 2008. [11] W. Imrich and P. F. Stadler, Minimal cycle bases of product graphs, Australasian J. of Combinatorics 26 (2002) 233-244. [12] P. C. Kainen, On robust cycle bases, Proc. 9th Quadr. Conf. in Graph Theory, June 2000, Y. Alavi et al., Ed., in Electr. Notes in Discr. Math. 11 (2002), 430-437 (art. 38, pp. 1-8). [13] P. C. Kainen, Isolated squares in hypercubes and robustness of commutativity, Cahiers de Topologie et Géométrie Différentielle Catégoriques XLIII (2002), 213-220. [14] P. C. Kainen, Graph cycles and diagram commutativity, pp. 177-238, in Diagrammes, Sup-plem. aux vol. 67+68, Paris, 2012. [15] K. Klemm and P. F. Stadler, Statistics of cycles in large networks, Phys. Rev. E 73 (2006) 025101. [16] K. Klemm and P. F. Stadler, A note on fundamental, non-fundamental, and robust cycle bases, Discrete Applied Mathematics 157 (2009), 2432-2438. [17] S. Mac Lane, Categories for the working mathematician, Springer, New York, 1971 [18] F. Orejas, On the representation of data types, in Formalization of Programming Concepts, Lecture Notes in Computer Science, Vol. 107, 1981, 419-431. [19] P-J Ostermeier, M. Hellmuth, K. Klemm, J. Leydold and P. F. Stadler, A note on quasi-robust cycle bases, Ars Mathematica Contemporanea 2 (2009), 231-240. [20] I. Peterin, A characterization of planar partial cubes, Discrete Math. 308(2008), 6596-6609. [21] B. C. Pierce, Basic Category Theory for Computer Scientists, MIT Press, 1991. [22] M. M. Syslo, On cycle bases of a graph, Networks 9 (1979), 123-132. [23] A. Weinstein, Groupoids: Unifying Internal and External Symmetry, Notices Amer. Math. Soc. 43 (1996), 744-752. [24] A. White, Graphs of Groups on Surfaces, Mathematics Studies 188, North-Holland, Amsterdam, 2001. /^creative ^commor Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 45-50 Comparing the irregularity and the total irregularity of graphs Darko Dimitrov Institut fur Informatik, Freie Universität Berlin, 14195 Berlin, Germany Riste Skrekovski * Department of Mathematics, University of Ljubljana, 1000 Ljubljana and Faculty of Information Studies, 8000 Novo Mesto, Slovenia Received 9 June 2012, accepted 15 August 2013, published online 13 June 2014 Abstract Albertson [4] has defined the irregularity of a simple undirected graph G as irr(G) = J2uvee(g) I dG(u) — dG(v) |, where dG(u) denotes the degree of a vertex u e V (G). Recently, in [1] a new measure of irregularity of a graph, so-called the total irregularity, was defined as irrt(G) = 2 J2u veV(G) |dG(u) - dG(v)|. Here, we compare the irregularity and the total irregularity of graphs. For a connected graph G with n vertices, we show that irrt (G) < n2irr(G)/4. Moreover, if G is a tree, then irrt(G) < (n — 2)irr(G). Keywords: The irregularity of graph, the total irregularity of graph. Math. Subj. Class.: 05C05, 05C07,05C99 1 Introduction Let G be a simple undirected graph of order n = |V(G)| and size m = |E(G)|. For v e V (G), the degree of v, denoted by dG ( v ), is the number of edges incident to v. Albert-son [4] defines the imbalance of an edge e = uv e E(G) asimbG(uv) = |dG(u) — dG(v)| and the irregularity of G as irr(G) = ^ imbG(uv). (1.1) uvEE(G) Obviously, a connected graph G has irregularity zero if and only if G is regular. In [4] Albertson presented upper bounds on irregularity for bipartite graphs, triangle-free graphs * Partially supported by Slovenian ARRS Program P1-00383 and Creative Core - FISNM - 3330-13-500033. E-mail addresses: dimdar@zedat.fu-berlin.de (Darko Dimitrov), skrekovski@gmail.com (Riste Skrekovski) and arbitrary graphs, as well as a sharp upper bound for trees. Some results about the irregularity of bipartite graphs are given in [4, 14]. Related to the work of Albertson is the work of Hansen and Melot [13], who characterized the graphs with n vertices and m edges with maximal irregularity. Various upper bounds on the irregularity of a graph were given in [19], where Kr+1-free graphs, trees and unicyclic graphs with fixed number of vertices of degree one were considered. In [16], relations between the irregularity and the matching number of trees and unicyclic graphs were investigated. More results on irregularity, imbalance and related measures, one can find in [3, 5, 6, 17, 18]. Recently, in [1] a new measure of irregularity of a simple undirected graph, so-called the total irregularity, was defined as irrt(G) = 2 E |dG(u) - do(v)|. (1.2) u,veV (G) Other approaches, that characterize how irregular a graph is, have been proposed [2, 3, 7, 8, 9, 10, 15]. In this paper, we focus on the relation between the irregularity (1.1) and the total irregularity (1.2) of a graph. In the sequel we introduce the notation used in the rest of the paper. For u, v E V (G), we denote by dG(u, v) the length of a shortest path in G between u and v. In this short paper the notation of the sets, that will be defined next, is always regarding the graph G we consider. By Va,b, we denote a set of vertices of a graph with degrees in [a, b], and by V>a (resp. Va (resp. V dT (x, v)}. Each summand \dG(u) — dG(v)\ in the last sum of (2.1) occurs in the sum exactly nuv = nunv times. Also, each summand \dG(v) - dG(u)\ occurs nuv times. Thus, irr t (G) - £ \dG(u) - dG(v)\nuv. uveE(T ) As nuv < (n/2)(n/2) = n74, and Euv£E(T) \dG(u) - dG(v) \ < EuveE(G) \dG(u) -dG (v) \, we obtain the desired inequality. Now, we show that the bound n2/4 is sharp. Let a, b be two distinct integers, say a < b. Consider a graph Ga whose all vertices are of degree a, with exception of one vertex u which is of degree a - 1. Similarly, consider a graph Gb whose all vertices are of degree b, with exception of one vertex u which is of degree b - 1. Let G* be the graph obtained from Ga and Gb by connecting u and v. Let na = \V(Ga)\ and nb = \V(Gb)\. Observe that irr(G*) = b - a and irrt(G*) = (b - a)nanb. Choosing na = nb = n/2, we obtain irrt(G*) n2 ■ = nanb = "j". irr(G* ) 4 In order to show that such graphs Ga and Gb exist, one may use the theorem of Erdos-Gallai [12] which states that a sequence d1 > d2 > • • • > dn of non-negative integers with even sum is graphic (i.e., there exist a graph with such a degree sequence) if and only if rn £dj < r(r - 1) + £ min(r, dj), (2.2) i=1 j=r+1 for all 1 — r — n. So, fix a, b, and na = nb to be odd numbers with na > max{a, b}. We will show the existence of the graph Ga. In a similar way, one can show the existence of the graph Gb. As (na - 1)a + (a - 1) is even, the parity condition of the theorem of Erdos-Gallai is satisfied. So, we need to show only (2.2). For this we consider three cases regarding r and a: • r — a - 1. Then, (2.2) can be written as ra — r(r - 1) + (na - r)r. It obviously holds since a C na - r. • r = a. In this case, (2.2) can be written as ra — r(r - 1) + (na - r)r - 1, which holds for a similar reason as the previous case. • r > a + 1. Similarly, (2.2) can be written as ra — r(r - 1) + (na - r)a - 1, and it holds as ra C r(r - 1). □ 3 Trees In this section, we give an upper bound on irrt(G) in term of irr(G), when G is a tree. To show the bound, we will use the following lemma. Lemma 3.1. Let G be a tree, x a vertex of degree d > 3 incident with threads T and T2, and let G' = G(T2 ◦ T1). Then, (a) irrt(G) - irrt(G') = 2v2,d~i; (b) irr(G) - irr(G') = 2(d - v>d - 1). Proof. Let T1 = a\a2 • • • as and T2 = b1b2 • • • bi. We consider the identities separately. (a) Notice that all other vertices except x and bl have the same degree in G and G'. Hence, it holds that irrt(G) - irrt(G') = ^ (|dG(x) - dG(u)|-|dG/ (x) - do' (u)|) u=bi + ^(|do(u) - do(bi)| - |do'(u) - do'(bl)|) u=x + |do(x) - do(bi)| - |do'(x) - do'(bi)|. Since d0'(x) = do(x) - 1 = d - 1 and d0'(bl) = do(bl) + 1 = 2, further we have irrt(G) - irrt(G') = ^ (|d - do(u)|-|d - 1 - do(u)|) u=bi + ^(|do(u) - 1|-|do(u) - 2|) + 2. (3.1) u=x If u G Vd. Similarly, if u G V>2, then |do(u) - 1| - |do(u) - 2| = 1, otherwise |do(u) - 1| -|do(u) - 2| = -1. Thus, the second sum in (3.1) is equal to v>2 - 1 - v1. Applying these observations, we have irrt(G) - irrt(G') = vd + v>2 - 1 - vi +2 = v2 - v>d = 2v2,d-i. (b) Let e1 = xa1, e2 = xb1, e3 = bl-1bl and E1 = {e1, e2, e3}. Denote by E2 the set of edges incident to x that are different from e1 and e2. Notice that every edge not in E1 U E2 contributes zero to the difference irr(G) - irr(G'). So, we can infer irr(G) - irr(G') = ^^ (imbo(uv) - imb0' (uv)) uvGE2 + ^ (imbo(uv) - imb0'(uv)). uvGEi Notice that the first sum is equal to -v|d + (v 2: In this case, imbG(e1) = d - 2 and imbG (e'J = 0. In both of them, we obtain imbG(e1) - imbG' (e1) = d - 2. Next, we consider the contributions of e2 and e3 together. Again, consider two possibilities regarding the length of T2: • l = 1: Then, e2 = e3 and imbG(e2) = d - 1 and imbG'(e2) = d - 3; • l > 2: In this case, e2 = e3, and imbG(e2) = d - 2, imbG(e2) = d - 3, imbG(e3 ) = 1 andimbG' (e3) = 0. In both cases, we obtain that J2ee{e2 e3}(imbG(e) - imbG'(e)) = 2. So finally, we have that irr(G) - irr(G') = -v|d + (v|d_1 - 2) + d - 2 + 2 = -v|d + vXXd-1 - 2 + d = 2(d - v|d - 1). □ Theorem 3.1. Let G be a tree with n vertices. Then irrt(G) < (n - 2)irr(G). Moroever, equality holds if and only if G is a path. Proof. Let n1 (G) be the number of vertices of G with degree one. We will prove the second inequality by induction on n1(G). If n1(G) = 0, then G ~ P1, irr(G) = irrt(G) = 0, and the equality in the theorem holds. Since G is a tree, n1 (G) = 1. If n1(G) = 2, then G ~ Pn. In this case irr(G) = 2 and irrt(G) = 2(n - 2), hence we obtain equality. Now, assume n1 (G) > 2. Then, it is easy to see that G has a vertex x of degree d > 3, incident with at least two threads T1 and T2. Let G' = G(T2 о T1). Since n1(G') = n1(G) - 1, we can assume that inequality holds for G', i.e., irrt(G') < (n - 2)irr(G'). (3.2) By Lemma 3.1, we have irr(G') = irr(G) - 2(d - v|d - 1) and irrt(G') = irrt(G) - 2v2,d-1. (3.3) Plugging (3.3) in (3.2), we obtain (n - 2)irr(G) > irrt(G) - 2v2,d-1 + 2(n - 2)(d - v|d - 1). (3.4) As d(x) = d > 3 and x is incident with two threads, we infer v|d + 2 < d, and so 2(d - v|d - 1) > 2. Observe also that v2,d-1 < n - 3. Hence 2(n-2)(d - v%d - 1) > 2(n - 3) > 2v2,d-1. This together with (3.4) gives (n - 2)irr(G) > irrt(G). □ References [1] H. Abdo, S. Brandt and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014), 201-206. [2] Y. Alavi, A. Boals, G. Chartrand, P. Erdos and O. R. Oellermann, k-path irregular graphs, Congr. Numer. 65 (1988) 201-210. [3] Y. Alavi, G. Chartrand, F. R. K. Chung, P. Erdos, R. L. Graham and O. R. Oellermann, Highly irregular graphs, J. Graph Theory 11 (1987) 235-249. [4] M. O. Albertson, The irregularity of a graph, Ars Comb. 46 (1997) 219-225. [5] F. K. Bell, A note on the irregularity of graphs, Linear Algebra Appl. 161 (1992) 45-54. [6] Y. Caro and R. Yuster, Graphs with large variance, Ars Comb. 57 (2000) 151-162. [7] G. Chartrand, P. Erdos and O. R. Oellermann, How to define an irregular graph, Coll. Math. J. 19 (1988) 36-42. [8] G. Chartrand, K. S. Holbert, O. R. Oellermann and H. C. Swart, F-degrees in graphs, Ars Comb. 24 (1987) 133-148. [9] L. Collatz and U. Sinogowitz, Spektren endlicher Graphen, Abh. Math. Sem. Univ. Hamburg 21 (1957) 63-77. [10] D. CvetkoviC and P. Rowlinson, On connected graphs with maximal index, Publications de l'Institut Mathematique (Beograd) 44 (1988) 29-34. [11] D. Dimitrov, T. Reti, Graphs with equal irregularity indices, Acta Polytech. Hung. 11 (2014), 41-57. [12] P. Erdos and T. Gallai, Graphs with prescribed degrees of vertices, (in Hungarian) Mat. Lapok. 11 (1960) 264-274. [13] P. Hansen and H. Melot, Variable neighborhood search for extremal graphs 9. Bounding the irregularity of a graph, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 69 (1962) 253-264. [14] M. A. Henning and D. Rautenbach, On the irregularity of bipartite graphs, Discrete Math. 307 (2007) 1467-1472. [15] D. E. Jackson and R. Entringer, Totally segregated graphs, Congress. Numer. 55 (1986) 159165. [16] W. Luo and B. Zhou, On the irregularity of trees and unicyclic graphs with given matching number, Util. Math. 83 (2010) 141-147. [17] D. Rautenbach and I. Schiermeyer, Extremal problems for imbalanced edges, Graphs Comb. 22 (2006) 103-111. [18] D. Rautenbach and L. Volkmann, How local irregularity gets global in a graph, J. Graph Theory 41 (2002) 18-23. [19] B. Zhou and W. Luo, On irregularity of graphs, Ars Combin. 88 (2008) 55-64. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 51-61 Maximum genus, connectivity, and Nebesky's Theorem Dan Archdeacon Department of Mathematics and Statistics, University of Vermont Burlington, VT, USA 05405 Michal Kotrbcik Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia Roman Nedela Mathematical Institute, Slovak Academy of Sciences, 975 49 Banskä Bystrica, Slovakia Martin Skoviera Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia Received 19 July 2012, accepted 10 October 2013, published online 20 June 2014 We prove lower bounds on the maximum genus of a graph in terms of its connectivity and Betti number (cycle rank). These bounds are tight for all possible values of edge-connectivity and vertex-connectivity and for both simple and non-simple graphs. The use of Nebesky's characterization of maximum genus gives us both shorter proofs and a description of extremal graphs. An additional application of our method shows that the maximum genus is almost additive over the edge cuts. Keywords: Maximum genus, Nebesky's theorem, Betti number, cycle rank, connectivity. Math. Subj. Class.: 05C10 1 Introduction We study cellular embeddings of graphs in orientable surfaces of large genus. Suppose that G is a connected graph with v vertices and e edges embedded with f faces on an orientable surface of genus g, denoted here by Sg. Our goal is to find the maximum genus of G, E-mail addresses: dan.archdeacon@uvm.edu (Dan Archdeacon), kotrbcik@dcs.fmph.uniba.sk (Michal Kotrbcik), nedela@savbb.sk (Roman Nedela), skoviera@dcs.fmph.uniba.sk (Martin Skoviera) Abstract YM (G), which is the maximum value of g over all cellular embeddings of G. The Euler-Poincaré formula asserts that v — e + f = 2 - 2g. By combining this formula with the formula e(G) = 1 — v + e for the Betti number (cycle rank) of G we get f — 1 = ß — 2g. Since the maximum genus g corresponds to the minimum number of faces f and since every embedding has at least one face, f — 1 (or equivalently ß — 2g) measures how far the embedding is from the extreme of only one face. We call this quantity the deficiency £ of the embedding. The deficiency of G, £(G), is the minimum deficiency over all possible embeddings, and so £(G) = ß(G) — 2ym (G). Note that £(G) always has the same parity as ß(G). In particular, an embedding of a graph with odd Betti number always has at least two faces and hence deficiency at least one. A graph with deficiency 0 or 1 is called upper embeddable. For more background on the maximum genus we refer the reader to [3]. In this paper we give lower bounds N on the maximum genus of graphs in various classes C defined by the graph's Betti number and edge-connectivity. The graphs are allowed to have loops and multiple edges; if these are forbidden we say that the graph is simple. Our results can be rephrased as "Every graph in C has an embedding in a surface with at least N handles". We also get analogous results for Betti number and vertex-connectivity. These classes have been examined before [2, 4, 5, 7, 8, 11], nevertheless, we present a unified approach with new, much shorter proofs, providing additional information about the structure of the extremal graphs. In addition to the bounds on the maximum genus of a graph in terms of its connectivity, we explore the relationship between the maximum genus of a graph and the maximum genera of components resulting from the removal of an edge-cut. By an edge-cut in a graph G we mean a minimal set F of edges that separate two sets of vertices V1 and V2 forming a partition {V1, V2} of the vertex-set of G; every edge from F has precisely one vertex in each of Vi and V2. We show that the maximum genus of a graph is almost additive over its edge-cuts (in a sense to be made clear in Section 3). We also investigate a related question asking for the smallest size of an edge-cut that can separate a non-upper-embeddable component. The primary tool used in this paper is Nebesky's Theorem [12], stated below as Theorem 1.1 (or its equivalent version due to Khomenko and Glukhov [9] stated later in Equation 3.4). Let G be a connected graph and let A be a set of its edges. Let oc (G — A) denote the number of components of G — A with odd Betti number and ec(G — A) the number of those with even Betti number. Let v(G, A) = ec(G — A) + 2oc(G — A) — |A| — 1. Theorem 1.1 (Nebesky [12]). The deficiency of a graph G is given by £(G) = max{v(G, A) : A C E (G)}. The idea behind Nebesky's Theorem is easy and its understanding is important to this paper. Consider a maximum-genus embedding of G with f faces. Delete the edges of A one at a time. If deleting an edge increases the number of components, then we embed each new component on its own connected surface. With this convention, deleting an edge can increase the total number of faces, but by at most one. Hence the embedding of G — A has at most f + |A| faces. If a component of G — A has odd Betti number, then its surface has at least two faces. If the Betti number is even, then its surface has at least one face. It follows that f + |A| > ec(G — A) + 2oc(G — A), and so £(G) = f — 1 > v (G, A). Nebesky's Theorem also asserts that there is a set A of edges in a maximum genus embedding that reverses the steps just described; the proof in this direction is more difficult. Previous results relating maximum genus, connectivity, and the Betti number mostly used Xuong's Theorem [15] to determine the maximum genus. It is our use of Nebesky's alternative characterization [12] that gives us new insights and shorter proofs; these techniques were first introduced in an earlier unpublished version of this paper (1996). 2 Graphs with given connectivity In this section we state and prove our results giving upper bounds on the deficiency of graphs with a particular Betti number and connectivity. We then translate these results to give lower bounds on the maximum genus. We need several observations before proceeding. First, note that if G has a vertex v of degree one, then ym (G - v) = ym (G). If v is of degree two, then we can replace its two incident edges with a single edge joining the other two incident vertices. This does not change the maximum genus, but can turn a simple graph into a non-simple one. Since simple and non-simple graphs behave differently, the statements of our results are cleaner if we forbid degree two vertices. Hence we require that our graphs are of minimum degree at least three. The proofs involve calculating v (G, A) and bounding ß(G) from below. Different types of components in G - A will play different roles. Let c0 = c0(G - A) be the number of components in G - A with Betti number zero. Let c1 denote those components with Betti number one. Let c2 be the number of components with even Betti number at least 2, and c3 be the number of components with odd Betti number at least 3. With this notation observe that v(G, A) = co + 2ci + C2 + 2c3 - |A| - 1. (2.1) To bound the Betti number of G, first note that G - A has components whose Betti numbers sum to at least c1 + 2c2 + 3c3. Adding in c0 + c1 + c2 + c3 - 1 edges from A, we can build a connected graph whose Betti number is also at least c1 + 2c2 + 3c3. Adding the remaining edges in A increases the Betti number by |A| - (c0 + c1 + c2 + c3 - 1). Hence ß(G) >-co + c2 + 2c3 + |A| + 1. (2.2) We are now ready for the first version of our main result. Theorem 2.1. Let G be a graph of minimum degree at least three. Then upper bounds on the deficiency £(G) are given in following table. The rows correspond to edge-connectivity k =1,2,3, or > 4. The same bounds hold where k is the vertex-connectivity and are achieved by graphs of arbitrarily large Betti number. k simple non-simple 1 2 3 > 4 (ß(G) - 2)/2 (ß(G) = 3) (ß(G) - 4)/3 (ß(G) = 3, 5) (ß(G) - 4)/3 (ß(G) = 3, 5) 1 ß(G) ß(G) - 2 (ß(G) - 4)/3 (ß(G) = 3, 5) 1 Proof. We prove the upper bounds for k-edge-connected graphs G. Since k-vertex-con-nected implies k-edge-connected, the bounds also hold for vertex-connectivity. We achieve the bounds with k-vertex-connected graphs, so the bounds are best possible in both cases. The proof has five main cases. Case (i). We begin with possibly non-simple graphs having edge connectivity (or vertex connectivity) k > 4. First observe that v (G, 0) is 0 when ß(G) is even and is 1 when ß(G) is odd. We will show that no 4-edge-connected G has a non-empty subset A with v (G, A) > 0. As a consequence, any such graph is upper-embeddable and the bounds in our table are best possible. This result has been noted by many authors, usually using Kundu's Theorem [10] combined with Xuong's Theorem [15]. To establish the inequality, first note that if A is non-empty and G-A is connected, then v (G, A) < 0. If G - A is disconnected, then each component of G - A is incident with at least4 edges in A. Counting edge ends and dividing by 2 we get |A| > 2c0+2ci+2c2+2c3. Substituting in Equation 2.1 gives v (G, A) < -c0 - c2 - 1 < 0 as desired. Case (ii). We next consider graphs that are either 3-edge-connected and non-simple or 2-edge-connected and simple. The bound for simple graphs was first shown in [2]. Let A be a non-empty subset of edges. We need to show that v(G, A) < (ß(G) - 4)/3. Substituting Equation 2.1 for v(G, A), Equation 2.2 for ß(G), and simplifying, it suffices to show that If G is 3-edge-connected, then every component is incident with at least three edge-ends from A. Hence 2|A| > 3(c0 + c1 + c2 + c3) implying Equation 2.3. The bound is achieved if and only if c0 = c2 = c3 = 0. If G is 2-edge-connected and simple, then every component of G - A contributing to c0 is incident with at least 3 edges (by minimum degree at least 3). A component contributing to c1 is incident with at least 3 edges by simplicity. The remaining components are incident with 2 edges by connectivity. Hence 2|A| > 3c0 + 3c1 + 2c2 + 2c3. Again, Equation 2.3 is shown and the bounds follow. The bound is achieved if and only if c0 = c2 = 0, every component contributing to c1 is a triangle and every component contributing to c3 is incident with precisely 2 edges from A. If A is an empty set of edges, then v(G, A) = 0 or 1, depending on whether e(G) is even or odd respectively. This is less than (e(G) - 4)/3 unless e(G) = 3 or 5, hence the two excluded cases. One way to achieve the above bound is to replace every vertex of a 3-connected simple graph with a triangle so that every vertex is of degree three. Here the set A corresponds to all of the original edges in the graph. This example shows that the bound is best possible in both classes. Case (iii). The next case is when G is simple and connected, first done in [4]. As before, we need to show that v(G, A) < (ß(G) - 2)/2 for every pair (G, A). Again substituting in Equations 2.1 and 2.2, it suffices to show that Our goal is to show that there is a pair (G, A) minimizing ^ with a certain structure, allowing us to show that ^(G, A) > 0. We start by choosing a pair (G, A) minimizing ^(G, A) where A is minimal with respect to inclusion, and proceed by proving several claims about (G, A). 2|A| > 2co + 3ci + c2 +2c3. (2.3) m(G, A) := 3|A| - (3co + 4ci + c2 + 2c3) + 1 > 0. (2.4) Claim 1: Every edge from A joins two distinct components of G - A. If there were an edge x such that both ends of x are incident with a single component of G — A, then p(G, A — {x}) < p(G, A), which contradicts the minimality of A. Claim 2: Each component of G — A has odd Betti number. To see this, first note that c0 = 0. If not, we could replace an acyclic component with a cycle of suitable length incident with the same edges of A as the original component to obtain a simple graph G' with minimum degree at least 3. For the resulting pair (G', A) we have p(G', A) < p(G, A), which contradicts the minimality of p. We further observe that c2 = 0. If not, we could replace a component contributing to c2 with a suitable subdivision of K4 contributing to c3. This can always be done in such a way that the resulting graph G' is simple and has minimum degree at least 3. However, p(G', A) < p(G, A), again contradicting the minimality of p. Claim 3: No component of G — A is incident with exactly two edges of A. Suppose there were such a component C, and let e, f denote the edges from A incident with C. As shown above, ß(C) is odd, and since G is simple and has minimum degree at least three, ß(C) > 3. Consider a pair (G', A') constructed by deleting C from G and joining the other ends of the edges incident with C. It is not difficult to see that G' has minimum degree at least three and that p(G', A') < p(G, A); however, the graph G' may not be simple. If the graph G' is simple, then p(G', A') < p(G, A) contradicts the minimality of p. If G' is not simple, then we distinguish two cases: e and f join C with either one or two components of G — A. If there is one component, it can contribute to p by at most 4. It follows that p(G, A — {e, f}) < p(G, A), contradicting the minimality of A. If there are two components call them D1 and D2. Then joining the ends of e and f in G creates a pair of parallel edges in G'. Moreover, this implies that Di and D2 are joined by some edge in A, say g. As both D1 and D2 can contribute to p by at most 4 and C contributes by 2, it follows that p(G, A — {e, f, g}) < p(G, A), again contradicting the minimality of A. Claim 4: A component C of G — A has e(C ) = 1 if and only if it is incident with at least three edges of A. Let C be a component of G — A incident with at least three edges. If ß(C) > 3, then we can replace it with a cycle of suitable length to obtain a simple graph G' with minimum degree at least 3. For the resulting pair (G', A') we again have p(G', A') < p(G, A), a contradiction. If D is a component incident with just one edge of A, then the simplicity and the minimum degree of G guarantee that ß(D) > 3. This proves Claim 4. Having established some properties of G — A, we return to the proof of Case (iii). Form the graph H = G/(G — A) from G by contracting each component of G — A into a vertex. Note that |E(H) | = |A| and that H has no vertices of degree 2. As H is connected, |E(H)| > c1 + c3 — 1. Denote the number of vertices of degree 1 in H by v1 and the number of vertices of degree at least 3 by v3. Observe 2|E(H)| > 3v3 + v1 = 3c1 + c3. By adding the two inequalities we get 3|E(H)| = 3|A| > 4c1 + 2c3 — 1. Substituting this into (2.4) yields p(G, A) = (4c1 + 2c3 — 1) — (4c1 + 2c3) + 1=0. To achieve the bound (ß(G) - 2)/2 > v(G, A), start with any tree H having only vertices of degree 1 or 3. Replace each degree three vertex with a triangle and each degree one vertex with a copy of K4 (possibly with one edge subdivided) so that every vertex has degree at least three. Case (iv). We next turn to the case where G is non-simple and 2-edge-connected. Every component is incident with at least two edges in A, so | A| > c0 + c1 + c2 + c3. This gives v(G, A) < ci + c3 - 1 and ß(G) > c1 + 2c2 + 3c3 + 1. To maximize v with an upper bounded ß we must have c2 = c3 = 0. Hence v (G, A) < ß(G) - 2, giving another entry in our table. This bound is achieved by the graph formed by replacing every other edge of the cycle C2n with two edges in parallel. These graphs are the necklaces of [2]. Case (v). When G is non-simple and 1-edge-connected we have |A| > c1 + c2 + c3 - 1. This gives v (G, A) < ß(G) as desired. The bound is achieved by duplicating every other edge on a path with an even number of edges and contracting one of the two edges incident with a vertex of degree two. Complete characterization of non-simple 1-edge-connected graphs achieving this bound can be found in [13]. We have completed filling in the table and the proof of our theorem. □ Recall that for a graph G embedded in the surface Sg, we have f - 1 = ß(G) - 2g, and hence £(G) = ß(G) - 27m (G). We use this formula to translate the upper bounds on £ to lower bounds on the maximum genus 7M , giving the following second form of our main result. Theorem 2.2. Let G be a graph of minimum degree at least three. Then lower bounds on the maximum 7M (G) are given in following table. The rows correspond to edge-connectivity k = 1,2,3, or > 4. The same bounds hold where k is the vertex-connectivity and are achieved by graphs of arbitrarily large Betti number. k simple non-simple 1 (ß(G) + 2)/4 (ß(G) = 3) 0 2 (ß(G) + 2)/3 (ß(G) = 3, 5) 1 3 (ß(G) + 2)/3 (ß(G) = 3, 5) (ß(G) + 2)/3 (ß(G) = 3, 5) > 4 (ß(G) - 1)/2 (ß(G) - 1)/2 □ We note that our table is slightly different than the one in [2] where, for example, they give the lower bound [ß (G)/3] for the maximum genus of 2-edge-connected simple graphs of minimum degree at least 3. Both bounds are tight. They are achieved only for graphs with ß(G) congruent to one modulo three, when the bounds are the same. 3 Edge-cuts We now investigate the relationship between the deficiency of a graph and the deficiency of components resulting from the removal of an arbitrary edge-cut. Our motivation is twofold. First, the assumptions concerning arbitrary edge-cuts are much weaker than the connectivity requirements used in Section 2. Second, similar ideas were pursued by Jaeger, Payan, and Xuong [6] who proved that, under natural parity conditions, a graph that can be separated by an edge-cut into two upper-embeddable components is itself upper-embeddable. Our aim is to generalize this result by replacing upper-embeddability with an arbitrary deficiency. The resulting lower bound for deficiency suggests the following question: "What is the minimum size of an edge-cut that can separate a non-upper-embeddable component in a к-edge-connected graph for к > 4?" Our final result, Theorem 3.4, answers this question. As in the previous section, a crucial role is played by various types of components depending on their Betti numbers. Let F be a fixed edge-cut of a connected graph G with components Gi and G2. Let A be an arbitrary subset of edges of G. We distinguish between two types of components of G - A: a pure component has its vertices entirely in G1 or entirely in G2, while a mixed component is incident with vertices in both. Let d0 denote the number of mixed components of G - A with even Betti number. Let d1 denote the number of pure components of G - A with even Betti number and with vertices from G1. Define d2 similarly for pure components with vertices from G2. Finally, let d3, d4, and d5 denote the odd counterparts of d0, d1, and d2, respectively. Notice that pure components contribute to d1,d2,d4, and d5 and mixed components contribute to d0 and d3. With this notation, Theorem 1.1 implies the following: £(G) > v(G,A) = ec(G - A) + 2oc(G - A) — |A| — 1 = do + d1 + d2 + 2d3 + 2d4 + 2d5 - |A| - 1. (3.1) We now proceed to the main result of this section, Theorem 3.1. To show that the bounds stated in it are tight we need one more ingredient, Xuong's Theorem [15]: The deficiency of a graph G equals the minimum number of components with odd number of edges (which we call odd for short) in the subgraph G - E (T), where the minimum is taken over all spanning trees T of G. Theorem 3.1. Let G be a connected graph and let F be an edge-cut such that G - F has precisely two components G1 and G2. Then Ì(G1) + £(G2) - |F| + 1 < £(G) < £(G1 ) + £(G2) + 1, both bounds being tight. Equivalently, Ym(G1) + YM(G2) + |F|/2 - 1 < YM(G) < YM(G1 ) + YM(G2) + |F| - 1. Proof. First we show that £(G) < £(G1) + £(G2) + 1. Let A be some set of edges of G maximal with respect to inclusion such that £(G) = v (G, A). Set Aj = A П E (Gj) for i G {1, 2} and note that A - (A1 U A2) = A n F. (3.2) Our goal is to bound the values of v(G1, A1) and v(G2, A2). Pure components are also components of either G1 - A1 or G2 - A2. Therefore, a pure component contributing to d1 or d2 increases the value of v(G1, A1 ) or v(G2, A2) by precisely 1. Pure components contributing to d4 or d5 increase the value of v(G1, A1) or v(G2, A2) by precisely 2. If D is a mixed component, then D - F has at least one component contained in G1 and at least one component contained in G2. Therefore D increases v(G1, A1) + v(G2, A2) by at least 2. Note that if ß(D) is even (that is, if D contributes to d0), then D increases v (G, A) only by one. It follows that ec(G1, A1) + 2oc(G1, A1) + ec(G2, A2) + 2oc(G2, A2) > 2d0 + d1 + d2 + 2d3 + 2d4 + 2d5. (3.3) Combining (3.1), (3.2), and (3.3) we get £(Gi) + £(G2) > v (Gì, Ai) + V(G2, Аз) = ec(G1 ,A1) + 2oc(G1,A1) - |A11- 1 + ec(G2,A2) + 2oc(G2,A2) - |Аз| - 1 > 2do + d1 + d2 + 2d3 + 2d4 + 24 — IA1I — |A2| — 2 = ec(G - A) + 2oc(G - A) + d0 - |A11 - |A21 - 2 = ec(G - A) + 2oc(G - A) - |A| - 1 + d0 + |A - (A1 U A2)| - 1 = e(G) - 1 + |F n A| + do > e(G) - 1. This establishes the upper bound on £(G). To prove the lower bound, for i g {1, 2} let Aj denote a set of edges of G i maximal with respect to inclusion such that v(Gi, Ai) = ^(Gi). Set A = A1U A2 U F and calculate: £(G) > v (G, A) = ec(G - A) + 2oc(G - A) - |A| - 1 = ec(G1 - A1) + ec(G2 - A2) + 2oc(G1 - A1) + 2oc(G2 - A2) — |A11 — |A21 - |F|- 1 = v(G1, A1) + v(G2, A2) - |F| + 1 = e(G1) + £(G2) - |F| + 1. This establishes the lower bound on £(G). To see that both bounds are tight, take two copies of the dipole Dn, which has two vertices and n parallel edges, and join the copies by two independent edges e and f. Let G be the resulting graph. Using Xuong's Theorem it is easy to verify that £(Dn) = 1 if n is even, £(Dn) = 0 if n is odd, and that £(G) = 1. Let F = {e, f} and let G1 and G2 be the components of G - F. Then £(G) = £^1) + £(G2) - |F| + 1 if n is even, and £(G)= C(G1) + C(G2) + 1 if n is odd. □ Theorem 3.1 and the fact that £(G) and ß(G) have the same parity give a shorter proof for the following result of Jaeger, Payan, and Xuong [6]. Corollary 3.2. Let G be a connected graph and let F be an edge-cut of G such that G — F has precisely two components G1 and G2, both upper-embeddable. Then G is upper-embeddable provided that both e(G1) and e(G2) are even, or precisely one of e(G1) and в(G2) is even and e(G) is odd. □ In the final part of the proof of Theorem 3.1 we have shown that the lower bound on £(G) is tight in the class of 2-edge-connected graphs. Our next example shows that the lower bound is tight also in the class of 3-edge-connected graphs. Example 3.3. We construct a rich infinite family of 3-edge-connected graphs G containing an edge-cut F whose removal produces two components G1 and G2 such that £(G) = C(G1) + £(G2) - |F| + 1. Recall that the truncation of a cubic graph H is a cubic graph t(H) obtained by expanding every vertex of H into a triangle. Let us start with connected cubic graphs H1 and H2 without loops, of order n1 and n2, respectively, both greater than 2. Take their truncations G1 = t(H1) and G2 = t(H2), and connect G1 to G2 by a set F = {f0, f1, f2} of three edges arbitrarily. Let G be the resulting graph; it is easy to see that if H1 and H2 are 3-edge-connected, so is G. Bouchet [1] proved that £(Gj) = Ui/2 - 1 (see [14, Theorem 3.3] for a different proof). We now show that £(G) = £(Gi) + £(G2) - 2 = e(Gi) + £(G2) - |F| + 1 = (ni + n2)/2 - 4. For i g {1, 2} let Ai be the set of edges of G i not lying in a triangle, and let A = Ai U F U A2. Since |A| = 3 + 3(ni + n2)/2 and G - A consists of ni + П2 triangles, Nebesky's Theorem implies that £(G) > v (G, A) = 2(ni + П2) - 3(ni + n2)/2 - 3 -1 = C(Gi) + £(G2) - 2. To prove the reverse inequality, we take in each Hi a spanning tree Si such that all components of Hi - E(Si) are paths; it is not difficult to see that this choice is indeed possible [14, Theorem 3.1]. Extend Si to a spanning tree Ti of Gi by including in Ti two edges from each triangle of Gi. This can be done in such a way that a component of Hi - E(Si), which is a path of length m > 0, becomes a path of length 2m +1 constituting a component of Gi - E(Ti). A straightforward calculation reveals that there are ni/2 - 1 such components in Gi - E (Ti), all other components being isolated vertices. In particular, Gi - E(Ti) has ni/2 -1 = ^(Gi) odd components. Form a spanning tree T of G by adding the edge f0 to Ti U T2. Clearly, T is also a spanning tree of G' = G - {fi, f2} and the corresponding cotree has (ni + n2)/2 - 2 = ^(Gi) + £(G2) = £(G') odd components. We now add fi and f2 to G' one by one and modify T, if necessary, each time absorbing one odd component. Pick f i and note that each of its end-vertices belongs to a component Pi of Gi - E (Ti) or a component P2 of G2 - E(T2). Clearly, Pi U {f i} U P2 is a component of (G' + fi ) -E (T ). If Pi and P2 have different parity, then Pi U {fi} U P2 is even, implying that the cotree (G' + fi) - E (T ) has £(Gi) + £(G2) - 1 odd components. If both Pi and P2 are odd, then Pi U{fi}U P2 is an odd component of (G' + fi) - E(T) and therefore the cotree (G' + fi) - E(T) has C(Gi) + £(G2) - 2 + 1 odd components. It remains to consider the case where Pi and P2 are both even. Let v be the end-vertex of fi in Gi, and let K be the triangle of Gi containing v. The construction of T\ implies that Pi coincides with v and that both edges of K incident with v belong to T\; take one of them, say k. The third edge of K, denoted by q, belongs to Gi-E(T\), and the component P of Gi-E(Ti) containing it is an odd path. Note that P - q consists of two even paths, possibly trivial. It follows that T ' = T + k - q is a spanning tree of G' + fi such that the component of (G' + fi) - E (T ') containing fi is no more odd. Hence, the cotree (G' + fi) - E(T') has ^(Gi) + £(G2) - 1 odd components, which in turn implies that £(G' + fi) < ^(Gi) + £(G2) - 1. To finish the proof that £(G) < £(Gi ) + £(G2 ) - 2 we add f2 to G' + f i and proceed similarly, except that we modify T2 instead of T\. ■ We next turn our attention to 4-edge-connected graphs. Our next result exhibits a dramatic change in the behavior with regard to the smallest size of an edge-cut that can separate a component with large deficiency: the size of an edge-cut that separates a component with deficiency m must be at least linear in m. A leaf of a graph is a 2-edge-connected subgraph maximal with respect to inclusion. For a graph H, let ol(H) denote number of leaves of H with odd Betti number. The following equivalent version of Nebesky's theorem holds for every connected graph G (see [9]): £(G) = max{ol(G - A) - |A| : A C E (G)}. (3.4) Theorem 3.4. Let G be a k-edge-connected graph with k > 4 and let F be an edge-cut of G whose removal produces a component with deficiency at least m. Then |F| > (k - 2)m + 2. Proof. Let H be a component of G - F with £(H) > m. Each edge of F has at most one end in H. Choose a subset A C E (H ) such that £(H ) = ol(H - A) - |A| and let l = ol(H - A). As G is k-edge-connected, every leaf from H - A must be incident with at least k edges in G. These edges can be only from F, from A, or they can be bridges of H - A. There are l - 1 bridges in H - A. It follows that kl < |F| +2(l - 1) + 2|A|. (3.5) By substituting l - £(H) for |A| in (3.5) and manipulating the resulting expression we derive |F| > 2£(H) + l(k - 4)+ 2. For k = 4 we get |F| > 2£(H) + 2 = (k - 2)£(H) +2 > (k - 2)m + 2, as required. If k > 5, then k - 4 > 1, and using the fact that l > £(H) we obtain |F| > 2£(H) + l(k - 4)+ 2 > 2£(H) + e(H)(k - 4)+ 2 > (k - 2)£(H) + 2 > (k - 2)m + 2. This again gives the required inequality. □ The following corollary follows directly from Theorem 3.4. Corollary 3.5. If G is a k-edge-connected graph with k > 4 and F is an edge cut of G such that a component of G - F is not upper-embeddable, then |F | > 2k - 2. □ Let G be a 4-edge-connected graph and let F be an edge-cut separating it into components Gi and G2. Assuming that m = £(Gi) > £(G2), Theorem 3.4 implies that |F| > (k - 2)m + 2 > 2m + 2. Therefore, ^(G1) + £(G2) - |F| + 1 is always negative and the lower bound in Theorem 3.1 cannot be achieved for 4-edge-connected graphs. Acknowledgements. The research reported in this paper was supported from the following sources. The second author was partially supported by UK/217/2012 and by VEGA 1/1005/12, the third author was partially supported by APVV-0223-10, and the fourth author was partially supported by APVV project ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation. The authors would like to thank Jianer Chen, Yuangqiu Huang, Saroja Kanchi, Deming Li, and Yanpei Liu for fruitful discussions, and an anonymous reviewer for helpful comments. References [1] A. Bouchet, Genre maximum d'un Д-graphe, in: Problémes combinatoires et théorie des graphes, Colloques Internat. C. N. R. S., Paris, 1978, 57-60. (in French) [2] J. Chen, D. Archdeacon and J. L. Gross, Maximum genus and connectivity, Discrete Math. 149 (1996), 19-29. [3] J. Chen and Y. Huang, Maximum genus, in: Topics in topological graph theory, 34-44, Encyclopedia Math. Appl. 128, Cambridge Univ. Press, Cambridge 2009. [4] J. Chen, S. P. Kanchi and J. L. 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Liu, A tight lower bound on the maximum genus of a 3-connected loopless multigraph, Appl. Math. J. Chinese Univ. Ser. B 15 (2000), 369-376. [12] L. Nebesky, A new characterization of the maximum genus of a graph, Czechoslovak Math. J. 31 (1981), 604-613. [13] E. A. Nordhaus, R. D. Ringeisen, B. M. Stewart, and A. T. White, A Kuratowski-type theorem for the maximum genus of a graph, J. Combin. Theory Ser. B 12 (1972), 260-267. [14] M. Skoviera, The decay number and the maximum genus of a graph, Math. Slovaca 42 (1992), 391-406. [15] N. H. Xuong, How to determine the maximum genus of a graph, J. Combin. Theory B 26 (1979), 217-225. ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 63-75 The automorphism groups of non-edge-transitive rose window graphs Edward Dobson * Department of Mathematics and Statistics, Mississippi State University PO Drawer MA Mississippi State, MS 39762 Istvan Kovacs , Stefko MiklaviC UP IAM and UP FAMNIT, University of Primorska Muzejski trg 2, 6000 Koper, Slovenia Received 2 August 2012, accepted 7 September 2012, published online 20 June 2014 In this paper, we will determine the full automorphism groups of rose window graphs that are not edge-transitive. As the full automorphism groups of edge-transitive rose window graphs have been determined, this will complete the problem of calculating the full automorphism group of rose window graphs. As a corollary, we determine which rose window graphs are vertex-transitive. Finally, we determine the isomorphism classes of non-edge-transitive rose window graphs. Keywords: Rose window graphs, automorphism group, isomorphism problem, vertex-transitive graph. Math. Subj. Class.: 05E18 1 Introduction Rose windows graphs are defined as follows (we are using the notation and terminology as in [18]). Definition 1.1. Let n be a positive integer and a,r e Zn (so arithmetic with a and r is done modulo n). The rose window graph Rn(a, r) is defined to be the graph with vertex set V = {Ai, B i : i e Zn} and four kinds of edges: • Ai Ai+i These edges are called rim edges. * Project sponsored by the National Security Agency under Grant Number H98230-11-1-0179. E-mail address: dobson@math.msstate.edu (Edward Dobson), istvan.kovacs@upr.si (Istvan Kovacs), stefko.miklavic@upr.si (Stefko Miklavic) Abstract • AiBi These edges are called in-spoke edges. • Ai+aBi These edges are called out-spoke edges. • BiBi+r These edges are called hub edges. Rose window graphs were introduced recently by Steve Wilson [18], whose initial motivation was concerned with determining which of these graphs are edge-transitive (and if so what is their full automorphism group), as well as which of these graphs are the underlying graph of a rotary map. He proposed four conjectures concerning questions that he was interested in, and subsequently all of the conjectures have been shown to be true. Edge-transitive rose window graphs were characterized in [5, Theorem 1.2], verifying [18, Conjecture 11] (a graph is edge-transitive if its automorphism group acts transitively on the set of edges). The full automorphism groups of edge-transitive rose window graphs was determined in [6, §3], verifying [18, Conjectures 3 and 5]. The rose window graphs which are the underlying graph of a rotary map were found in [6, Theorem 1.1], answering [18, Question 3], Finally, [18, Conjecture 6] suggesting certain rose window graphs are isomorphic was verified in [6, Theorem 3.6]. Our goal is to essentially complete the work that has already been done regarding calculating the full automorphism groups of rose window graphs, as well determining exactly when two rose window graphs are isomorphic. In this paper, we will calculate the full automorphism groups of rose window graphs that are not edge-transitive (which will finish the problem of determining the full automorphism groups of rose window graphs), see Corollary 3.5 and Corollary 3.9. In Section 4, we will determine the isomorphism classes of rose window graphs that are not edge-transitive. The conclusion of the isomorphism problem for rose window graphs will be given in a sequel to this paper, where the isomorphism classes of edge-transitive rose window graphs will be found. There are a few additional results in this paper that should be mentioned. First, in Lemma 2.2, we correct a small error in [18, Lemma 2] giving conditions on when a rose window graph has an automorphism that maps every rim edge to a hub edge and vice versa. Also, once the full automorphism groups of rose windows graphs are known, it is relatively straightforward to determine which of these graphs are vertex-transitive, and this is given in Theorem 3.10. We should point out that our goal is a classical one. Namely, with graphs that have a large amount of symmetry, it is quite standard to ask for their full automorphism groups as well as their isomorphism classes. Perhaps the first family for which this has been done are the generalized Petersen graphs. The automorphism groups of these graphs were obtained by Frucht, Graver, and Watkins [4] in 1971, while the isomorphism classes were found by by Boben, Pisanski, and Žitnik [1]. Using very differrent techniques, Steimle and Staton [16] also determined the isomorphism classes for some, but not all, generalized Petersen graphs, and then used that result to enumerate the generalized Petersen graphs whose isomorphism classes they determined. Using Boben, Pisanski, and Žitnik's determination of the isomorphism classes of all generalized Petersen graphs, Petkovsek and Žakrajsek [9] enumerated generalized Petersen graphs. Determining the isomorphism classes of the rose window graphs should also yield an enumeration of the rose window graphs using techniques similar to those in [9]. Finally, in the last decade or so there has been considerable interest in tetravalent graphs satisfying various properties or in studying certain families of such graphs (for a sample of such work see [3, 7, 10, 11, 12, 13, 17]). See also [14], where a census of all locally imprimitive tetravalent arc-transitive graphs on up to 640 vertices was computed. This work will certainly contribute to the understanding of such graphs. 2 Preliminary results We first give some obvious automorphisms of rose window graphs. Let Rn(a, r) be a fixed rose window graph and let G be the automorphism group of Rn(a, r). Observe that Rn(a, r) = Rn(a, -r). (2.1) Define p, p : V ^ V by p(Ai)= Ai+1 and p(Bi) = Bi+1 (i G Zn), (2.2) p(Ai) = A-i and p(Bi) = B-a-i (i G Zn). (2.3) Note that p,p G G, and therefore (p, p) < G. The action of (p, p) on the set of edges of Rn(a, r) has three orbits: the set of rim edges, the set of hub edges and the set of spoke edges. The following result characterizes edge-transitive rose window graphs in terms of rim and spoke edges (we remark that the full automorphism groups of edge-transitive rose window graphs are given in [5], but the following formulation is nonetheless useful). Lemma 2.1. The following are equivalent: (i) Rn(a, r) is edge-transitive. (ii) There is an automorphism of Rn(a, r) which sends a rim edge to a spoke edge. (iii) There is an automorphism of Rn(a, r) which sends a spoke edge to a hub edge. Proof. It is clear that (i) implies (ii). To show that (ii) implies (iii), suppose that AiAi+1 is a rim edge mapped to a spoke edge by, say, a G G. Then a(AiAi+1) = AjBk for some j, к G Zn, and a(Ai) = Bk for I = i or i +1. Of course, e1 = A^B^ and e2 = A^B^-a are spoke edges, and a(e1) and a(e2) are two edges incident with the spoke edge AjBk, and all three of these edges are incident with a(A£) = Bk. However, Bk is incident with two hub edges and two spoke edges, so at least one of a(e1) and a(e2) must be a hub edge. To show (iii) implies (i), recall that the hub, spoke and rim edges are the edge orbits of (p, p). If a maps some spoke edge to a hub edge, we have that H = (p, p, a) has at most two edge orbits, and if there are two edge orbits, these consist of spoke and hub edges in one orbit and rim edges being the other orbit. However, if the rim edges form an orbit, then H must map {Ai : i G Zn} to itself, and so must map {Bi : i G Zn} to itself, and so must map hub edges to themselves. This then implies that H has three edge orbits, a contradiction. So H has one edge orbit and Rn (r, a) is edge-transitive. □ It follows that if Rn(a, r) is not edge-transitive, then it has either two orbits or three orbits on edges. If Rn(a, r) has two orbits on edges, then one orbit consists of rim and hub edges, and the other consists of spoke edges. If Rn(a, r) has three orbits on edges, then the first one consists of rim edges, the second one consists of hub edges, and the third one consists of spoke edges. Lemma 2 in [18] states that there is an automorphism of Rn(a, r) sending rim edges to hub edges and vice-versa if and only if r2 = ±1 (mod n) and ra = ±a (mod n). However, this is not entirely true. Namely, one can check that the rose window graph R16 (8,3) has an automorphism sending rim edges to hub edges and vice-versa via the map (i, j) ^ (i, 11 j). However it is clear that r2 =9 ф ±1 (mod 16). We now wish to give a correct statement of [18, Lemma 2], and begin with a preliminary lemma. Lemma 2.2. Let a be the automorphism of Rn(a, r), which sends every rim edge to a hub edge and vice versa. Assume also that a(A0) = B0 and a(B0) = A0. Then one of the following holds for every i e Zn: (i) a (Ai) = Bri and a(Bi) = Ari; (ii) a(Ai) = Bri anda(Bi) = A(r+a)i; (iii) a(Ai) = B-ri and a (Bi) = A-ri; (iv) a(Ai) = B-ri and a(Bi) = A(_r+a)i Proof. Since a(A0) = B0 and a(A1) are adjacent, we have a(A1) e {Br,B-r}. It is easy to see that if a(A1) = Br then a(Ai) = Bri for i e Zn, and that if a(A1) = B-r then a(Ai) = B-ri for i e Zn. Now let s e Zn be such that a(B1) = As and note that a(Bi) = Asi for i e Zn. Moreover, a(A1) and a(B1) = As are adjacent. Therefore, if a(A1) = Br, then s e {r, r + a}, and if a(A1) = B-r, then s e {— r, —r + a}. The result follows. □ Theorem 2.3. Let n > 3 bean integer and a,r e Zn \ {0}. Then there is an automorphism of Rn (a, r) sending every rim edge to a hub edge and vice-versa if and only if one of the following holds: (i) a = n/2, r2 ф 1 (mod n) and ra ф ±a (mod n); (ii) a = n/2, r2 ф ±1 (mod n) and ra ф ±a (mod n); (iii) n is divisible by 4, gcd(n, r) = 1, a = n/2 and (r2 + n/2) ф ±1 (mod n). Proof. We first show that if (i), (ii) or (iii) holds, then there is an automorphism of Rn(a, r) sending rim edges to hub edges and vice-versa. By (2.1) we can assume that ra ф —a (mod n). Observe that if one of conditions (i), (ii) or (iii) holds, then gcd(n, r) = 1. If condition (i) or (ii) holds, then let a : V ^ V be defined by a(Ai) = Bri and a(Bi) = Ari. As gcd(n, r) = 1, a is a bijection. It is straightforward to check that a is also an automorphism of Rn(a, r). If condition (iii) holds, then let a : V ^ V be defined by a(Ai) = Bri and a(Bi) = Ari+(n/2)i. Let us show that a is a bijection. As gcd(n, r) = 1, a maps {Ai | i e Zn} to {Bi | i e Zn} bijectively. As gcd(n, r) = 1, r is odd and n/2 is even, a maps {Bi | i e Zn} to {Ai | i e Zn} bijectively. Hence a is a bijection. It is then straightforward to check that a is also an automorphism of Rn(a, r). We now show that if there is an automorphism a of Rn(a,r) sending rim edges to hub edges and vice-versa, then either (i), (ii) or (iii) holds. Note that in this case it must be the case that gcd(n, r) = 1. Since (p, p) acts transitively on the sets of hub, rim and spoke edges, we may assume (by replacing a by an appropriate element of (p, p, a)) that a(A0) = B0 and a(B0) = A0. Using (2.1) we can further assume that a(A1) = Br. Therefore, by Lemma 2.2, a(Ai) = Bri and a(Bi) = Asi for i e Zn, where s e {r, r+a}. Since a2 sends A0 to A0 and A1 to Ars, we have rs ф ±1 (mod n). Consider an in-spoke AiBi and an out-spoke BiAi+a. The automorphism a maps the in-spoke AiBi to BriAsi, and the out-spoke BiAi+a to AsiBri+ra. Hence one of BriAsi and AsiBri+ra is an in-spoke, and the other one is an out-spoke. Therefore, for every i e Zn either ri = si (mod n) and ri + ra + a = si (mod n) (2.4) or ri + ra = si (mod n) and ri + a = si (mod n). (2.5) Note that if (2.5) holds for i = 0, then a = 0, a contradiction. Therefore (2.4) holds for i = 0, implying ra = —a (mod n). If (2.4) holds for i = 1, then we have r = s. Since rs = ±1 (mod n) this implies r2 = ±1 (mod n). If a = n/2, then (ii) holds. If a = n/2, then multiplying the congruence ra = —a (mod n) by r, we obtain r2a = —ar = a (mod n). If r2 = —1 (mod n), then —a = a (mod n). This implies that a = n/2, a contradiction. So if a = n/2, then r2 = 1 (mod n). Thus (i) holds. Suppose now that (2.5) holds for i = 1. Then r + ra = s (mod n) and r + a = s (mod n). The two congruences then imply that ra = a (mod n) and r + a = s (mod n). Since also ra = —a (mod n), we have that a = n/2 and r is odd. Combining together r + n/2 = s (mod n) and rs = ±1 (mod n) gives us (r2 + n/2) = ±1 (mod n). Observe that a(Bn/2) = As(n/2) = A(r+n/2)(n/2). Suppose n is not divisible by 4. As r and n/2 are both odd in this case, we have a(Bn/2) = A0 = a(B0). But this implies that a is not a bijection, a contradiction. Therefore, condition (iii) holds. □ It follows from Theorem 2.3 that Rn(a, r) has two orbits of edges if and only if one of conditions (i) or (ii) in Theorem 2.3 is satisfied. We will also use the following result. Lemma 2.4. Assume that Rn(a, r) is not edge-transitive and a = n/2. Then at least one of (i) r2 = ±1 (mod n) (ii) r2 + n/2 = ±1 (mod n) does not hold. Proof. If both (i) and (ii) above hold, then n/2 is congruent to 2, 0 or —2 modulo n. But this is only possible if n = 4. If n = 4, then r e {1, 3}. In both cases Rn(a, r) is edge transitive, a contradiction. □ 3 Groups of non edge-transitive rose window graphs Before proceeding, we will require some additional notation. Let N = gcd(n, r) denote the number of inner cycles, and let L = n/N denote the length of an inner cycle. Here an inner cycle is a cycle induced by some set of vertices {Bi | i e Zn}. We now define three types of permutations on V(Rn(r, a)). To do this we assume that n is even. For 0 < I < n/2 — 1, we define a permutation on V(Rn(r, a)) by a.£ = (B£, Be+n/2). If L is even, then for 0 < i < N — 1 we let ße — (Be, Be+n/2)(Be+N, Be+N+n/2)(B£+2W, Be+2N+n/2) • • • (Be+n/2-N; Be+n-N). Observe that ße interchanges every two antipodal vertices of the inner cycle containing Be. If L is odd, then for 0 < t < N/2 - 1 we let Ye = (Be+o,Be+n/2)(Be+N,Be+N+n/2)(Be+2N,Be+2N+n/2) ••• (Be+n-N; Be+n-N+n/2). Observe that Ye interchanges the inner cycle containing Be and the inner cycle containing Be+n/2. Lemma 3.1. Assume n is even. Then the following hold: (i) For 0 < t < n/2 — 1 we have ae — pea0p-e. (ii) If L is even, then for 0 < t < N — 1 we have ße — peß0p-e. (iii) If L is odd, then for 0 < t < N/2 — 1 we have Ye — peY0p-e. Proof. (i) It is straightforward to check that (pea0p-e)(Ai) — Ai for every i e Zn and that (pea0p-e)(Bi) — Bi for every i e Zn \{t,t + n/2}. Similarly we find that pea0p-e interchanges Be and Be+n/2. The result follows. (ii) Since ß0 — a0aNa2n • • • an/2-n and ße — aeae+Nae+2N • • • ae+n/2-N the result follows from (i) above. (iii) Similarly as (ii) above. □ Lemma 3.2. Assume n is even. Then the following hold: (i) If L — 4 then ae is an automorphism of Rn(n/2, r) for 0 < t < n/2 — 1. (ii) If L is even, L — 4, then ße is an automorphism of Rn(n/2, r) for 0 < t < N — 1. (iii) If L is odd then Ye is an automorphism of Rn(n/2, r) for 0 < t < N/2 — 1. Proof. Straightforward. □ Lemma 3.3. Let G a be the point-wise stabiliser of {A0, Ai,..., An-i} in G. Then the following hold: (i) If a — n/2 then G a is trivial. (ii) If a — n/2 and L — 4, then G a — (a0, a1,..., an/2-1). (iii) If a — n/2, L is even and L — 4, then G a — (ß0, ß1,..., ßN-1). (iv) If a — n/2 and L is odd, then G a — (y0, Yb ..., YN/2-1). Proof. Let a e G a. Since the outer cycle (that is, the n-cycle induced by the vertices {Ai | i e Zn}) is fixed by a, for every i e Zn we have either a(Bi) — Bi and a(Bi-a) — Bi-a, or a(Bi) — Bi-a and a(Bi-a) — Bi. If a is nontrivial, then there exists j e Zn such that a(Bj) — Bj-a and a(Bj-a) — Bj. Applying the above comment to i — j + a we find that a(Bj+a) — Bj and a(Bj) — Bj+a. Therefore j — a — j + a, implying a — n/2. This proves (i). Assume L = 4. Every a (1 < £ < n/2 — 1) is clearly in Ga by Lemma 3.2 (i). Therefore (a0,a\,... ,an/2-1) < G a. Pick a e Ga. Since for every i (i e Zn) the automorphism a either fixes or interchanges Bi and Bi+n/2, we clearly have G a < (a0, ai,..., «n/2-i). Therefore G a = («0, ai,..., «n/2-i). Assume L is even, L = 4. By Lemma 3.2 (ii), (ß0, ß1,..., ßN-1) < Ga. Pick a e Ga. For every i (i e Zn) the automorphism a either fixes or interchanges Bi and Bi+n/2. However, since L = 4, if a interchanges Bi and Bi+n/2, then it must interchange every pair of antipodal vertices of the inner cycle containing Bi (and therefore also Bi+n/2). Hence G a < (во, ßi,..., ßN-i), implying Ga = (во, ßi,..., ßN-i). Assume L is odd. Again, by Lemma 3.2 (iii), we have (j0, y1,..., yn/2-1) < G a. Pick a e Ga and assume that a interchanges Bi and Bi+n/2. Note that Bi and Bi+n/2 are now in different inner cycles. Therefore, a must interchange every Bj of the inner cycle containing Bi with Bj+n/2 (which is contained in the same inner cycle as Bi+n/2). It is now clear that a e (70, Yi,..., yn/ 2-i). This implies G a = (yo, Yi,..., Yn/2-i). □ Proposition 3.4. Let G{a} be the set-wise stabiliser of {A0, Ai,..., An-i} in G. Then the following hold. (i) If a = n/2 then G{ A} = (p, rf). (ii) If a = n/2 and L = 4, then G{a} = (p, M,a0 ). (iii) If a = n/2, L is even and L = 4, then G{a} = (p, V, ß0). (iv) If a = n/2 and L is odd, then G{a} = (p, M, Yo ). Proof. Let a e G{a}. Observe that the group induced by G{a} on A is (p, p), since the subgraph induced by A is a cycle. Therefore, pk^a e Ga for appropriate k e Zn, £ e Z2. The result now follows from Lemma 3.3 and Lemma 3.1. □ Corollary 3.5. Assume the automorphism group of Rn(a, r) has three orbits on the edge-set of Rn(a, r) (that is, Rn(a, r) does not satisfy any of the conditions (i) and(ii) of Theorem 2.3). Then the following hold. (i) If a = n/2 then G = (p, у). (ii) If a = n/2 and L = 4, then G = (p, ^, a0). (iii) If a = n/2, L is even and L = 4, then G = (p, ^,ß0 ). (iv) If a = n/2 and L is odd, then G = (p, ^, y0). Proof. If Rn(a, r) has three orbits on the edge-set, then one of these three orbits is the set of rim edges. Therefore G = G{a}. The result now follows from Proposition 3.4. □ We now turn our attention to the case when Rn(a, r) has two orbits on edges. In this case, in view of Lemma 2.1, the rim edges and the hub edges are in the same orbit, implying that gcd(n, r) = 1. Additionally, every automorphism of such a rose window graph must either fix the rim and hub or interchange them. Now suppose that we have an automorphism ш that interchanges the rim and hub. For any automorphism S of Rn(a, r), we then have that uS or S is contained in G a (noting that the set-wise stabilizer of {Ai : i e Zn} is the same as the set-wise stabilizer of {Bi : i e Zn}). Thus in order to calculate the automorphism groups of such graphs, we need only find one ш that interchanges the rim and hub, and then G = (Ga, w). Of course, Ga is given in Lemma 3.3, and we need only consider the parameters listed in Theorem 2.3. Definition 3.6. (i) Assume r2 = ±1 (mod n) and ra = —a (mod n). Then we define S : V ^ V by S(Ai) = Bri and 6(Б,) = Ari. (ii) Assume n is divisible by 4, r is odd, a = n/2 and (r2 + n/2) = ±1 (mod n). Then we define 7 : V ^ V by y(Aì) = Bri and y(Bì) = AH+(n/2)i. Lemma 3.7. Assume r2 = ±1 (mod n) and ra = —a (mod n). Then S G G, where S is as defined in Definition 3.6(i). Proof. Note that S is a bijection since gcd(n, r) = 1. The proof of the fact that S is an automorphism of Rn(a, r) is straightforward. □ Lemma 3.8. Assume n is divisible by 4, r is odd, a = n/2 and (r2 + n/2) = ±1 (mod n). Then 7 G G, where 7 is as defined in Definition 3.6(ii). Proof. We will show that 7 is a bijection as once this is established it is straightforward to verify that 7 G G. Clearly, 7 maps {Ai | i G Zn} bijectively to {Bi | i G Zn} as r is a unit. Similarly, 7 maps {Bi | i G Zn, i odd} bijectively to {Ai : i G Zn, i odd}, and {Bi | i G Zn, i even} to {Ai : i G Zn, i even}. Hence 7 is a bijection. □ Corollary 3.9. Assume the automorphism group of Rn(a, r) has two orbits on the edge-set of Rn(a, r). Then, in view of Theorem 2.3, the following hold. (i) If a = n/2 and r2 = 1 (mod n), then G = (p, p, S). (ii) If a = n/2, r2 = ±1 (mod n) and ra = —a (mod n), then G = (p, p, во, S). (iii) If n is divisible by 4, r is odd, a = n/2 and (r2 + n/2) = ±1 (mod n), then G = (P,P,e0,7). We remark that in the case (ii) of the previous corollary when r2 = —1, listing ß0 as a generator is redundantas S2p = ß0. In (iii), ß0 is superfluous as if r2 + n/2 = —1 (mod n) then ß0 = 72p while if r2 + n/2 = 1 (mod n) then ß0 = p-17pr7. The full automorphism group of all rose window graphs are now known with the previous result. We may then check each case to determine which are vertex-transitive. But given that p is always an automorphism of Rn(a, r), Rn(a, r) is vertex-transitive if and only if there is an automorphism of Rn(a, r) which maps a rim vertex to a hub vertex and an automorphism which maps a hub vertex to a rim vertex. Recall that a rose window graph has either three, two or one edge orbit. It has at most two edge orbits if and only if there is an automorphism which maps rim edges (vertices) to hub edges (vertices) and vice versa. Therefore, a rose window graph is vertex-transitive if and only if it has either one or two edge orbits. The edge-transitive rose window graphs are given in [5] and their full automorphism groups were obtained in [6]. Combining this information with Theorem 2.3, we obtain the following result which characterizes exactly which rose window graphs are vertex-transitive. Theorem 3.10. Let n > 3 be an integer and a, r G Zn \ {0}. The rose window graph Rn(a, r) is vertex-transitive if and only if one of the following holds: (i) r2 = ±1 (mod n) and ra = ±a (mod n); (ii) n is divisible by 4, r is odd, a = n/2 and (r2 + n/2) = ±1 (mod n); (iii) n is divisible by 2, a = n/2 ± 2, and r = n/2 ± 1; (iv) n is divisible by 12, a = ±(n/4 + 2), and r = ±(n/4 — 1) or a = ±(n/4 — 2) and r = ±(n/4 + 1); or (v) n is divisible by 2, a = 2b, where b2 = ±1 (mod n/2), and r is odd such that r = ±1 (mod n/2). 4 Isomorphisms of non edge-transitive rose window graphs Let Rn(a, r) and Rn(a1, r1 ) be non edge-transitive rose window graphs. In this section we consider the problem of finding conditions on a, r, a1, r1 to ensure that Rn(a, r) and Rn(a1,r1) are isomorphic. For the remainder of this paper, we will, as usual, denote the vertices of Rn(a, r) by {A0, A1,..., An-1} U {B0, B1,..., Bn-1}. The vertices of the rose window graph Rn(a1,r1) will be denoted in the natural way by {C0, C1,..., Cn-1}U {D0, D1,..., Dn-1}. Let p and ^ denote the automorphisms of Rn(a, r) defined at the beginning of this paper, and p1 and the corresponding automorphisms of Rn(a1,r1). Theorem 4.1. Let Rn(a, r) and Rn(a1, r1) be rose window graphs. If one of the following holds, then Rn(a, r) and Rn(a1,r1) are isomorphic: (i) r1 = ±r and a1 = ±a; (ii) gcd(n,r) = 1, r1 = ±r-1, and a1 = ±ar-1; (iii) n is even with gcd(n, r) = gcd(n, n/2 + r), a = a1 = n/2 and r1 = ±(r + n/2); (iv) n is even with gcd(n, r) = gcd(n, n/2 + r) = 1, a = a1 = n/2 and r1 = ±(r + n/2)-1; (v) r = ±1, r1 = ±1, gcd(n, a) = gcd(n, a1) = 2 and aa1/2 = ±2(mod n); (vi) gcd(n, n/2—1) = 1, r = ±(n/2—1), r1 = ±(n/2—1), gcd(n, a) = gcd(n, a1) = 2 and aa1/2 = ±2(mod n). Proof. (i) Note that Rn (a, r ) = Rn (a, —r ) and that an isomorphism between Rn (a, r ) and Rn(—a, r) is given by ^(Ai) = C-i and ^(Bi) = D-i for i e Zn. (ii) Assume gcd(n, r) = 1, r1 = r-1, and a1 = ar-1. Then an isomorphism from Rn(a, r) to Rn(a1, r1) is given by ^(Ai) = D-ir-i and ^(Bi) = C-ir-i for i e Zn. The result now follows from (i) above. (iii) Let L = gcdnn,r) = gcd(n,n/2+r), the length of the inner cycles of Rn(a, r) and Rn(a1, r1). We first claim that L is divisible by 4. To this end let n = 2ino and r = 2jro, where no and ro are odd positive integers. Since gcd(n, r) = gcd(n, n/2 + r), we also have that gcd(n, r) = gcd(n/2, r), and so j < i — 1. Assume now that j = i — 1. Then n/2 + r = 2i-1(no + ro) = 2i(no + ro)/2 (note that no + ro is even). This shows that gcd(n, n/2 + r) is divisible by 2i. Since gcd(n, r) is not divisible by 2i, we have a contradiction. Therefore, j < i — 2, and so L is divisible by 4. Now define ф : V(Rn(n/2, r)) ^ V(Rn(n/2, n/2 + r)) by ^(Ai) = Ci for i e Zn and ф^+ь-) = D£+fcr+fcn/2 for 0 < I < gcd(n, r) — 1 and 0 < k < L — 1. Choose an inner cycle C of Rn(n/2, r). Note that ф maps C to an inner cycle of Rn(n/2, n/2 + r), and while doing so, the only change in every other vertex is changing Bi to Di and on the remaining vertices ф interchanges "antipodal vertices" of the cycle. This will produce a bijection if and only if L is divisible by 4, and so ф is a bijection. It is now routine to check that ф is an isomorphism. The result now follows from (i) above. (iv) Immediately from (ii) and (iii) above. (v) Assume r =1, ri = 1, gcd(n, a) = gcd(n, a{) = 2 and aai/2 = 2 (mod n). Define a mapping from V(Rn(a,r)) to V(Rn(auri)) by ф(А2^ = C^, ф(A2i+l) = Dia!, = Cia, +1, Ф(B2i+l) = Dia1+i for 0 < i < n/2 — 1. Observe that ф is a bijection as gcd(n, a1) = 2. It is also clear that ф is an isomorphism. The result now follows from (i) above. (vi) Assume r = n/2 — 1, r1 = n/2 — 1, gcd(n, a) = gcd(n, a1) = 2 and aa1/2 = 2 (mod n). Note that since gcd(n, n/2 — 1) = 1, we have that n/2 is even. Furthermore, since gcd(n, a) = gcd(n, a1) = 2, a/2 and a1/2 are odd. Define a mapping from V(Rn(a,r)) to V(Rn(abri)) by ф^) = C^, ф^т) = Diai, ф^) = C1+iai, ф(B2i+n/2+1) = D1+iai for 0 < i < n/2 — 1. Observe that ф is a bijection as gcd(n, a1) = 2. Furthermore, since a1/2 is odd (and so (n/4)a1 = n/2), ф is an isomorphism. The result now follows from (i) above. □ Theorem 4.2. Let ф : Rn(a, r) ^ Rn(a1, r1) be an isomorphism which sends every rim edge of Rn(a, r) to a rim edge of Rn(a1, r1 ). Then one of the following holds: (i) ai = ±a and ri = ±r; (ii) n is even with gcd(n, r) = gcd(n, r + n/2), a = a1 = n/2 and r1 = ±(r + n/2). Proof. Note that there exist k g Zn and I g {0,1} such that ^jpf ф maps vertex Ai to vertex Ci for each i g Zn. Therefore without loss of generality we can assume that ф maps vertex Ai to vertex Ci for each i g Zn. Observe also that ф maps the hub (spoke) edges of Rn(a, r) to the hub (spoke) edges of Rn(a1, r1). It follows that ф^о) G {D0, D-ai}. Claim 1: If ф^0) = D0 then a1 = a. If, in addition, a = n/2, then r1 = ±r. Assume ф^0) = D0. Since vertices B0 and Aa are adjacent, vertices ф(B0) = D0 and ф(Aa) = Ca are also adjacent. Since a = 0 this shows that a1 = a. Assume a = n/2. As Ar and Br are adjacent, ф^г) = Cr and ф(Br ) are also adjacent. This shows that ф^г ) g {Dr, Dr-a}. As Ar+a and Br are adjacent, ф(Ar+а) = Cr+a and ф(Br) are also adjacent. This shows that ф^) g {Dr, Dr+a}. Note that since a G {0, n/2}, we have {Dr,Dr-a} n {Dr,Dr+a} = {Dr}. Therefore ф(Br) = Dr and r1 = ±r. This proves Claim 1. Claim 2: If ф^0) = D-ai then a1 = —a. If, in addition, a = n/2, then r1 = ±r. Rearranging the subscripts of the vertices {D0, D1,..., Dn-1} according to the rule x ^ x + a1 we get the graph Rn(—a1, r1) instead of the graph Rn(a1, r1). Furthermore, ф : Rn(a, r) ^ Rn(—a1,r1) now maps vertex B0 to (the new) vertex D0. By Claim 1 we have —ai = a and, if a = n/2, ri = ±r. This proves Claim 2. Assume now a = a1 = n/2 and r1 = ±r. Similarly as above, we find ф^0) G {D0,Dn/2}. If ф(Bо) = D0, then ф(Br) G {D,i,D_n} n {Dr,Dr+n/2}. Since ri = ±r, we have r1 = ±(r+n/2). It is clear that we have gcd(n, r) = gcd(n, r1) = gcd(n, r+ n/2). The case ф^о) = Dn/2 is treated similarly. □ Theorem 4.3. Let ф : Rn(a, r) ^ Rn(a1, r1) be an isomorphism which sends every rim edge of Rn(a, r) to a hub edge of Rn(a1, r1). Then one of the following holds: (i) a1 = ±ar-1 and r1 = ±r-1; (ii) n is even with gcd(n, r) = gcd(n, r + n/2) = 1, a = a1 = n/2 and ri = ±(r + n/2)-1. Proof. Since ф sends the rim edges of Rn (a, r) to the hub edges of Rn(a1,r1), it also sends the hub edges of Rn(a, r) to the rim edges of Rn(a1,r1). This shows that gcd(n, r) = gcd(n, r1) = 1. Rearranging the vertices of Rn(a1,r1) according to the rule Ci ^ Dir-1 and Di ^ Cir-1 for i G Zn we obtain the graph Rn(-a1r-1,r-1) instead of the graph Rn(a1,r1). Moreover, ф now satisfies the assumptions of Theorem 4.2. If Theorem 4.2 (i) holds, then r1 = ±r-1 and a1 = ±ar-1. If Theorem 4.2 (ii) holds, then n is even with gcd(n, r + n/2) = gcd(n, r) = 1, a = —a1r-1 = n/2 and r-1 = ±(r + n/2). Since r1 is odd (recall that gcd(n, r1) = 1), —a1r-1 = n/2 is equivalent to a1 = n/2. The result follows. □ Theorem 4.4. Let ф : Rn(a, r) ^ Rn(a1, r1) be an isomorphism which sends every rim edge of Rn(a, r) to a spoke edge of Rn(a1,r1). Then one of the following holds: (i) r = ±1, r1 = ±1, gcd(n, a) = gcd(n, a1) = 2 and aa1/2 = ±2 (mod n); (ii) gcd(n, n/2—1) = 1, r = ±(n/2—1), r1 = ±(n/2—1), gcd(n, a) = gcd(n, a1) = 2 and aa1/2 = ±2 (mod n). Proof. Observe first that as the rim edges of Rn(a, r) are mapped to the spoke edges of Rn(a1, r1), the outer cycle of Rn(a, r) is mapped to a cycle of even length in Rn(a1, r1). This shows that n is even. Next, as the rim edges of Rn (a, r ) are mapped to the spoke edges of Rn(a1, r1), the image of a rim edge has endpoints a rim vertex and a hub vertex. As hub edges and rim edges have no endpoints in common, the images of hub edges and rim edges have no endpoints in common. This implies that hub edges cannot be mapped either to the hub edges or the rim edges, and so the hub edges of Rn (a, r) are also mapped to the spoke edges of Rn(a1, r1). As the spoke edges of Rn (a1, r1) form a single edge orbit, we have that the hub and rim edges of Rn(a, r) also forms a single edge orbit. This shows that R„(a, r) and Rn(a1, r1) have two orbits on edges (and so gcd(n, r) = gcd(n, r1) = 1). We may thus assume that ф(А0) = Ci and ф(А1) g {Di,Di-ai} for some i g Zn. Multiplying ф by appropriate powers of and p1 we can further assume that ф(А0) = C0 and ф(А1) = D0. This implies that ф(А^) = Ciai and ф(Ак+1) = Diai for 0 < i < n/2 — 1. Therefore, the order of a1 in Zn is n/2 and thus gcd(n, a1) = 2. Reversing the role of Rn(a, r) and Rn(a1, r1) we also obtain that gcd(n, a) = 2. Note that this also shows that n, a and a1 are all even. Since R4(2,1) is edge-transitive, we may assume that n > 6. Observe now that ф(В0) g {C1,C-1}. We will assume ф(В0) = C1 ; the case ф(В0) = C-1 is treated similarly. Since B0 and Aa are adjacent, ф(В0) = C1 and ф(Аа) = C(a/2)ai are also adjacent. This shows that aa1/2 = 2(mod n). Furthermore, ф(Й2^) G {Co,C1,...,Cn_1} and ф(В(2ту) G {Do,D1,...,Dn_1} for 0 < i < n/2 — 1. Recall that gcd(n, r) = gcd(n, r1) = 1. In particular, r, r1,r-1 and r-1 are odd. Since B1 = Br-ir this shows that ф(В1) G {D0, D1,..., Dn-1}. Since B1 and A1 are adjacent, ф(В1) and ф(А1) = D0 are also adjacent. Since B1 and Aa+1 are adjacent, ф(В1) and ф(А^) = D(a/2)ai = D2 are also adjacent. Therefore ф(В1) G {Dri ,D-ri} n {D2+ri ,D2-ri}. This shows that r1 = ±1 or r1 = ±(n/2 — 1). Reversing the role of Rn(a, r) and Rn(a1, r1) we also obtain that r = ±1 or r = ±(n/2 — 1). Since Rn(a,r) = Rn(a, —r), we need only show r = 1 and r i = n/2 — 1 or r = n/2 — 1 and r1 = 1 cannot occur. Suppose that r = 1 and r1 = n/2 — 1. We saw in the previous paragraph that ф(В1) e {Dri, D_ri} n {D2+ri, D2-ri}. Since n > 6 this implies ф(В1) = Dn/2+1. But B0 and B1 are adjacent, and so ф(В0) = C1 and ф(В1) = Dn/2+1 are also adjacent. As 1 = n/2 +1 this implies n/2 +1 + a1 = 1 and thus a1 = n/2. It follows that gcd(n, a1 ) = n/2 > 3, contradicting gcd(n, a1) = 2. Finally, if r = n/2 — 1 and r1 = 1, then by reversing the roles of Rn(a,r) and Rn (a1, r1 ), this case cannot occur by arguments in the previous case. □ References [1] M. Boben, T. Pisanski and A. Žitnik, I-graphs and the corresponding configurations, J. Combin. Des. 13 (2005), 406-424. [2] E. Dobson and D. Witte, Transitive permutation groups of prime-squared degree, J. Algebraic Combin. 16 (2002), 43-69. [3] Y.-Q. Feng, J. H. Kwak, M.-Y. Xu, and J.-X. Žhou, Tetravalent half-arc-transitive graphs of order p4, European J. Combin. 29 (2008), 555-567. [4] R. Frucht, J. E. Graver and M. E. Watkins, The groups of the generalized Petersen graphs, Proc. 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ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 77-91 Minimal covers of equivelar toroidal maps Kostiantyn Drach * V.N. Karazin Kharkiv National University, School of Mathematics and Mechanics, Geometry Department, Svobody Sq. 4, 61022, Kharkiv, Ukraine Mark Mixer Wentworth Institute of Technology, Department of Applied Mathematics 550 Huntington Ave., Boston, MA 02115 Received 8 November 2012, accepted 29 April 2014, published online 11 July 2014 Given any equivelar map on the torus, it is natural to consider its covering maps. The most basic of these coverings are finite toroidal maps or infinite tessellations of the Euclidean plane. In this paper, we prove that each equivelar map on the torus has a unique minimal toroidal rotary cover and also a unique minimal toroidal regular cover. That is to say, of all the toroidal rotary (or regular) maps covering a given map, there is a unique smallest. Furthermore, using the Gaussian and Eisenstein integers, we construct these covers explicitly. Keywords: Minimal covers, Regular and rotary maps, Gaussian and Eisenstein integers. Math. Subj. Class.: 52B15, 51M20, 52C22 1 Introduction The classification of regular and chiral maps on the torus was first published in 1948 by Coxeter [3]. Recently, other classifications of toroidal maps have been presented, e.g. [1], [6], [11]. It is known that every equivelar toroidal map has a universal covering, in that it can be covered by a regular tessellation of the Euclidean plane. However, equivelar toroidal maps can be covered by smaller maps that also have high degrees of symmetry. Notably, each equivelar toroidal map has a unique finite minimal regular cover (see [7], [14]). Recently, there has been much interest in finding such minimal regular covers for different families of maps and abstract polytopes (see for example [8, 9, 17]). Our paper is part of this broader project of understanding minimal covers. However, we focus not only * Partially supported by the Akhiezer Foundation and I. E. Tarapov Scholarship. E-mail addresses: drach@karazin.ua (Kostiantyn Drach), mixerm@wit.edu (Mark Mixer) Abstract on regular covers, where much is known, but also on rotary covers (those that are either regular or chiral). In this paper we give a proof that any equivelar toroidal map has a unique minimal regular cover on the torus. While this result follows from the previously mentioned work on, our techniques allow for the explicit construction of these covers using the Gaussian and Eisenstein integers. Furthermore, we prove a similar result, that each such map is uniquely covered by a finite minimal rotary toroidal map, and again using number theoretic results, we can construct these covers. Our main results are the summarized by the following theorem. Theorem 1.1. Each equivelar map on the torus has a unique minimal rotary cover on the torus, and a unique minimal regular cover on the torus. Each can be constructed explicitly. An unpublished construction of these minimal regular covers was also made by Maksym Skoryk using an application of the theory of linear Diophantine equations. However, in this paper, we use a different approach, which yields the construction of minimal regular covers. Additionally, our construction provides the more general result concerning minimal rotary covers. The paper is organized as follows. In section 2 we give necessary definitions and state results about maps and covers. Section 3 consists mainly of relevant number theoretic results regarding the Gaussian and the Eisenstein integers. This section concludes with the technical statement of two theorems, which together prove Theorem 1.1. The following two sections provide the proof of these two theorems, where Section 4 deals with minimal rotary covers and Section 5 considers regular covers. Finally, in order to illustrate our results, we construct such minimal covers for some families of equivelar toroidal maps in Section 6. 2 Maps and Covers In this section we provide some definitions and results that will be of use in the future; most of these ideas, as well as further details, can be found in [2, 13, 18]. A finite graph X embedded on a compact 2-dimensional manifold S such that every connected component of S\X (which is called a face) is homeomorphic to an open disc is called a map M (on the surface S). An automorphism of M is an automorphism of the underlying graph X that can be extended to a homeomorphism of the surface S. The automorphisms of a map M form a group, which is called the automorphism group, denoted Aut(M ). Note that although any map is symmetric under uncountably many homeomor-phisms for the underlying surface S, we adopt a purely combinatorial (hence finite) point of view. A map M is equivelar of (Schlafli) type {p, q} if all its vertices are q-valent and all its faces are topological p-gons. Definition 2.1. Let M and N be maps on the surfaces S and S' respectively. A surjective function n : S' ^ S that preserves adjacency and sends vertices to vertices, edges to edges, and faces to faces of the maps M and N is called a covering of the map M by the map N. This is denoted by N \ M, and we say that N is a cover of M. It follows from Theorem 4.14 of [15], that any equivelar map M is covered by the universal map U of the same type {p, q}, and that M is a quotient of U by some subgroup of the automorphism group of U. If M and N are both of type {p, q} then, it can be easily shown from the above definition that there is a constant number K (possibly infinite) of faces of N that are sent to any face of M. This follows directly from the connectedness of the maps and the adjacency preserving property of a covering. Furthermore, this number K is also the number of edges or vertices of N that are sent to an edge or vertex of M. In this case, we say that N \ M is a K-sheeted covering. Hereafter, we consider equivelar maps of type {p, q} on a 2-dimensional torus T2, which are simply called equivelar toroidal maps. On the torus the only possible types to consider are {4,4}, {3, 6}, and {6, 3}. However, since any map of type {6, 3} is dual to one of type {3, 6}, and a cover of the dual is the dual of a cover, we will only consider maps of type {4,4}, and {3,6} in this paper. Equivelar toroidal maps can be seen as quotients of regular tessellations of the Euclidean plane. Given a regular tessellation т of the plane, we denote the group of translations that preserve the tessellation as TT. For each group TT, we can pick two shortest linearly independent translations ei, e2 which generate TT. Furthermore, in this paper ei, e2 are chosen with equal length and with the angle between them equal to П2 for {4,4}, or П for {3,6}. We call (e1, e2) the basis connected with the tessellation т. Hereafter, for each tessellation we fix the basis connected with it and will consider coordinates of all other vectors in the plane with respect to this basis. The next theorem gives an essential characterization of all equivelar toroidal maps. Theorem 2.2. [16] Let M be an equivelar map on the torus. Then M can be obtained as a quotient of a regular tessellation т of the Euclidean plane by some translation subgroup G - ì \ i Figure 1: {4,4}u,v \ {4,4}a,b is a 5-sheeted covering, and the covering map {4,4}u,v is obtained by gluing together 5 fundamental regions of {4,4}a,b Definition 2.6. [9], [13] An equivelar toroidal map N is a minimal rotary (regular) toroidal cover of an equivelar toroidal map M if N is a rotary (regular) map, N \ M, and for any other rotary (regular) equivelar toroid L such that N \ L \ M it follows that L = N or L = M. 3 The Gaussian and Eisenstein integers The Gaussian and Eisenstein integers provide an essential tool for constructing minimal covers for equivelar toroidal maps. Plotting these domains in the complex plane produces precisely the vertex set of a regular tessellation: т = {4,4} for the Gaussian integers and t = {3, 6} for the Eisenstein integers. Since we use the Gaussian and Eisenstein integers to construct minimal covering maps, we must recall some of their properties in this section. We will follow [12]. The Gaussian integers Z[i] are defined as {a + bi | a,b e Z}, where i = %/—1. Similarly, the Eisenstein integers Z[w] are defined as {a + bw | a,b e Z}, where w = (1 + ìa/3)/2. Hereafter, we write Z [a], a = i,w, to denote either of these two sets. In this paper we use the following notation: given a = a + ba e Z [a], we call its conjugate the number a := a + ba, where a is the conjugate complex number to a e C. Also we call Re a := a and Im a := b the real and imaginary parts respectively. Note here that if a = i, then this is the traditional notion of 'real part', and 'imaginary part' of a complex number. However, if a = w, then the 'traditional' real and imaginary parts of a + bw are a + b/2 and bV3/2, respectively. It is straightforward to prove the following technical lemma, which we will need later. Lemma 3.1. For every a,ß e Z[a] the following holds: (1) Im(aß) = Re a Im ß — Im a Re ß; (2) Im(aß) = — Im(aß). For every a = a + ba e Z[a], we assign the norm N (a) := aa. We recall that this is a multiplicative function; namely, for a, ß e Z[a], N (aß) = N (a)N (ß). A number a e Z[a]\{0} divides ß e Z[a] if and only if there is y e Z [a] such that ß = aj. In this case we will use the notation a|ß. Recall that, in the ring of Gaussian integers, the units are only ±1, ±i, while in the ring of Eisenstein integers the units are only ±1, ±w, ±w. Two numbers a, ß e Z[a] are associated (a ~ ß) iff a = ße for some unit e. It is easy to show that ~ is an equivalence relation. A number п e Z [a] is a prime in the ring Z [a] if and only if it is not a unit and a|n implies that a is either a unit or associated with п. Equivalently, п e Z [a] is a prime if and only if it is not a unit and from п| (aß) it follows that n|a or n|ß. The Gaussian and the Eisenstein integers are unique factorization domains and thus there is a theorem analogous to the fundamental theorem of arithmetic for these rings. Theorem 3.2 (Unique Factorization Theorem, [12]). Let Z[a] be the ring of Gaussian or Eisenstein integers and let PCT С Z[a] be a set of primes in Z[a] such that every prime number п e Z[a] is associated with a unique prime П e PCT, so that none of the numbers in PCT is associated with any other. Then for every a e Z [a], a = 0 there exists a unique representation: a = en^1 . .. where e is a unit, n1,... ,nn e PCT are pairwise distinct prime numbers, and e1,... ,en e N are natural numbers. Additionally, it is possible to describe all the primes in Z[a] in terms of their norms. For the Gaussian integers Z[i] there are the following possible types of primes n: (1) п ~ 1 + i, N(n) = 2. Since 1 + i = i(1 + i), n is associated with n; (2) rational prime numbers q = 3 (mod 4). As q G Z its conjugate q = q, q is associated with q; (3) n G Z[i] such that N (n) = p is an rational prime p = 1 (mod 4). Note that in this case П is a prime which is not associated with n. For the Eisenstein ring Z[w] there are similar types of primes: (1) n — 1 + ш; N(n) = 3, П — n; (2) rational prime numbers q = 2 (mod 3), q = q; (3) n G Z [ш] such that N (n) = p is an rational prime p = 1 (mod 3). n is a prime not associated with n. The following two lemmas follow from the Unique Factorization Theorem and the classification of the primes. The proofs for the both of this lemmas are straightforward and, therefore, omitted. Lemma 3.3. Let a G Z[a]\{0} and a = n00 qi1 ... qSs n^1 ... ntbt be its prime decomposition, where no — 1 + a, for j G {1,... s} the numbers qj are rational primes of type (2), for k G {1,... t} the numbers are primes of type (3), and all exponents are nonnegative integers. Then the following are equivalent: (1) GCD(Re a, Im a) = 1; (2) a0 G {0,1}, ai = ... = as =0 and nj — for all k,j G {1,..., t}. Lemma 3.4. Let a G Z[a]\{0}. Then (1+ a) |a if and only if Re a = Im a (mod N (1 + a)). A complex number 7 G Z [a] is a greatest common divisor (GCD) of a, ß G Z [a], N (a) + N (в) = 0, if Y|a and and for every 7' G Z [a] such that 7' |a and 7'\ß it follows y'|y. From the Unique Factorization Theorem it follows that a greatest common divisor is well defined up to associates. Thus, we can consider the GCD equivalence class [GCD(a, ß)]. Further, we use the common notation 7 = GCD(a,ß) implying that 7 G [GCD(a, ß)]. If there is arational integer n G [GCD(a, ß)], then we specifically take GCD(a, ß ) := |n|. For example, [GCD(3,6i)] = {3, -3, 3i, -3i}, and thus GCD(3, 6i) = 3. As we have seen, either the Gaussian or the Eisenstein integers can be linked with a regular tessellation т. Then the numbers 1 and a from Z[a] correspond to the basis (ei, e2 ) of translations connected with т, and the operations of addition and multiplication by an integer in Z [a] can be seen as the same operations in TT treated as a vector space over Z with the basis (e1, e2 ). From this point of view, we will identify every two vectors a = (ai, a2 ), b = (bi, b2) with the complex numbers a := ai + a2a G Z[a] and ß := bi + b2a G Z[a]. Finally, using this idea, we will denote та,в := та,ь. Furthermore we let тп := . It follows from Theorem 2.5 that, if n G Z[a]\{0}, then тп is a rotary toroidal map, and all rotary maps can be obtained in this way. Note that if n — n', then тп = тп>. This implies that we need only look for a generator n of a rotary map тп up to associates. Additionally, a rotary map тп is regular iff n — n or n — n(1 + a), where n G Z\{0}. Using the notation developed in this section, we are now ready to state the formal versions of the two theorems which together yield the main theorem of this paper. The proofs of these two theorems are the content of the next two sections. We remind the reader of our convenient but non-standard definition for Im 7 when 7 g Z[w]. Theorem 3.5. Let Ta,ß be an equivelar toroidal map represented as a quotient of a regular planar tessellation т by a translation subgroup (a, ß) < TT generated by two non-collinear vectors, corresponding to the complex numbers a,ß G Z[a]. Let y = GCD(a, ß). There exists a unique minimal rotary covering map equal to TVmin with Im (aß) Vmin = N(7) Y Moreover, the number Kmin of fundamental regions of Ta,ß glued together in order to obtain тп . is equal to K = |Im(aß)| Kmin = N (7) • Theorem 3.6. Let Ta,ß be an equivelar toroidal map represented as a quotient of a regular planar tessellation т by a translation subgroup (a, ß) < TT generated by two non-collinear vectors, corresponding to the complex numbers a, ß G Z [a]. Let c = GCD(Re a, Im a, Re ß, Im ß). Then for Ta,ß there exists a unique minimal regular covering map TVmin with Im(aß\(1+ a), if ^ = Ma and ^ = ^ mod N(1 + a); Im(eaß), otherwise• Moreover, the number Kmin of fundamental regions of Ta,ß glued together in order to obtain TVmin is equal to NmagL, ifBe. = Imea and Rß = ^ mod N(1 + a); K= min | Im(aß) | e2 otherwise. 4 Proof of Theorem 3.5 In this section we give the proof of our first main theorem, which states that every equivelar map on the torus has a unique minimal rotary cover on the torus. To accomplish this proof, we translate the concepts of covers from Section 2 into the terminology developed in Section 3, and construct all possible toroidal rotary covers for a given map. By doing so, we can explicitly find a cover that is minimal. Let Ta,ß be an equivelar toroidal map represented as a quotient of a regular planar tessellation t by a translation subgroup (a, ß) < TT generated by two non-collinear vectors, corresponding to complex numbers a, ß G Z [a]. We have seen that each rotary map can be described as тп for some n G Z [a]. Additionally, we have seen that covering maps satisfy the conditions of (2.1). Thus, in order to obtain a rotary map тп which covers Ta,ß, we need to find a number n G Z[a]\{0} such that: {ni a + mi ß = an, 1 " (4.1) n2a + m2ß = n, for some integers ni, mi, n2, m2, with n\m2 = n2m\. Let v = —ni + n2a and p = mi — m2a. we must find v, p G Z [a], N (v ) + N (p) = 0, where (using the notation of Section 3): П I — a Re v + ß Re u = an, { P " (4.2) a Im v — ß Im u = n We proceed by solving system (4.2) considering n as a parameter. From the possible solutions to this system, we can determine the exact value of n which generates the minimal cover rVmin \ Ta,ß. Multiplying the second equation by a, and subtracting the first from it; we get the following equation av — ßu = 0. (4.3) Let y := GCD(a,ß ). Then a/Y,ß/Y € Z [a], and GCD(a/Y, ß/Y) = 1. From this and (4.3) it follows that v = ß5, u = ^5, (4.4) YY for some 5 € Z[a]\{0}. In order to also be a solution to system (4.2) it is sufficient that Y|n, and thus n/Y € Z[a]. To solve system (4.2) we simply substitute (4.4) into the second equation from (4.2). This yields: a Im (ß-5)- ß Im (a5 Y Y Y Y Y After multiplying by 5 and taking the imaginary and real parts of both sides, we can use both parts of Lemma 3.1 to get that: Im ( -5 I =0, Therefore, Rei ?') = «(51 CmNgß) /Im(aß)\ - n = ЫлГ) 5Y. (4.5) Thus, v = ß/Y • 5 and u = й/y • 5 is a solution to system (4.2), and a rotary map тп covers ra,ß if and only if the parameter n satisfies (4.5). Notice here that ^ff € Z. Let us show that the unique minimal rotary cover is generated by the complex number (up to associates) Im (aß ) Vmin = «(Y) Y. Indeed, observe that for any 5i, 52 € Z[a]\{0} we have ts1s2 \ . Therefore, for any 5 € Z[a]\{0} and the corresponding complex number n satisfying (4.5), the map тп covers Trimin. Then from definition (2.6) it follows that rVmm is a minimal cover of та,в. Even more, if 5 is not a unit, then the corresponding rotary map тп = rVmin. Thus, rVmin is a unique minimal cover. The first part of Theorem 3.5 is proved. Let us find the number Kmin of fundamental regions of ra,ß that we should glue together in order to obtain rVmin. From (2.2), for an arbitrary cover тп substituting our Y solutions for p and v, this number is equal to K = |min2 — m2ni| = |Re p Im v — Re v Im p| = |Im (pv) Im ( a-ses |Im (ä.ß)l N(S). (4.6) Л Y / N (Y) ' Finally, since for the minimal cover - — 1, we obtain K = |Im (äß) Kmin = N (7) • This completes the proof of Theorem 3.5. 5 Proof of Theorem 3.6 In this section we give the proof of our second main theorem, which describes how every equivelar map on the torus has a unique minimal regular cover on the torus. Let ra,ß be an equivelar toroidal map represented as a quotient of a regular planar tessellation т by a translation subgroup (a, ß) < TT generated by two non-collinear vectors, corresponding to complex numbers a, ß G Z [а]. It follows from Theorem 2.5 that in order to produce any regular cover on the torus тп \ та,в we need to take a quotient of a regular tessellation т by a translation subgroup (n, an) < T b > 0, we have the following four cases. Example 6.1. For a map {4,4}(aj0),(o,b), we get a = a, ß = bi, 7 = GCD(a, bi) = GCD(a, b) = c, |Im(aß)| = ab. Thus the minimal regular (and rotary) cover is {4,4} ab, which can be obtained by gluing Ог fundamental regions of the initial map. Example 6.2. For a map {4,4}(ajb)j(aj_b), we get a = a + bi, ß = a — bi, c = GCD(a, b), |Im(aß)| = 2ab. Since a = ß, then 7 = c if a ф Г (mod 2), or 7 = c(1 + i) if а ф С (mod 2). In the first case the minimal regular (and rotary) cover is {4,4} 2ab, and the number of fundamental regions is equal to ^ In the second case the minimal regular (and rotary) cover is {4,4} ab (1+i), where the number of fundamental regions of the initial toroidal map we glued together in order to obtain this cover is equal to ф. Example 6.3. For a map {4,4}(a,b),(b,a) the situation is similar to Example 6.2. We have a = a + bi, ß = b + ai, c = GCD(a, b), and |Im(aß)| = a2 — b2. Since ia = ß, we get that y = c if а ф b (mod 2), or 7 = c(1 + i) otherwise. Thus, the minimal regular (and rotary) cover is built in the same manner as in Example 6.2. Example 6.4. For the final family of 2-orbit equivelar toroidal maps {4,4}(aa)(b_b), we get a = a + ai, ß = b — bi, c = GCD(a, b), 7 = c(1 — i), and |Im(aß)| = 2ab. Thus, the minimal regular (and rotary) cover is {4,4} ab(1+i), which can be obtained by gluing Or fundamental regions of the initial map (see Figure 2 (b)). Given a 4-orbit toroidal map, the minimal regular cover and the minimal rotary cover may or may not coincide; this depends on 7. For example, the minimal regular and minimal rotary covers of the map {4,4}(21) (7 0) coincide. On the other hand, the covers of {4,4}(4,3),(5,o) = {4,4}(4,3),(_i,3) are distinct (see Figure 2 (a)). 6.2 Maps of type {3, 6} The situation for regular and chiral toroidal maps of type {3, 6} is similar to the {4,4} case. Let us instead consider the families of 3-orbit maps of type {3, 6}. Let a,b e N. Example 6.5. For a 3-orbit toroidal map {3, 6}(a 0) (_b 2b), we have that a = a, ß = —b + 2Ьш, c = GCD(a, b), and |Im(aß)| = 2ab. Since ß = bw(1 + ш) , we know that 7 = c if O Ф 0 (mod 3) and 7 = c(1 + ш) otherwise. This implies that the minimal regular (and rotary) cover is equal to {3, 6} 2ab, where the number of fundamental regions glued to obtain the cover is 2Сзг in the first case; and {3, 6}^(1+ш), where the number of fundamental regions is equal to ЗаГ, in the second case. Example 6.6. For a 3-orbit map {3, 6}(a b) (a+b _b), we write that a = a + Ьш, ß = a + b — Ьш, c = GCD(a, b), and |Im(aß)| = 2ab + b2. Since a = ß, we get that 7 = c if f ф Г (mod 3) and 7 = c(1 + ш) otherwise. Thus, the minimal regular (and rotary) cover is {3,6} 2ab+b2, where the number of fundamental regions is equal to 2aC+г2 in the first case; and {3,6} 2ab+b^ ч, where the number of fundamental regions is 2a3b'+b2 in —зг— (1+ш) 3С the second case (see Figure 3(b)). As in the last subsection, the 6-orbit toroidal maps of type {3,6} have minimal regular and minimal rotary covers which may or may not coincide. Again this depends on 7 (see Figure 3). i n 2 ■ ■'I L Л \ 1 / / / у ■4 Л / a 3! / \ // / S / П2 2 / * n 1-- 4 / / a * / / 4 \ i ? ч / \ / 4 / (a) (b) Figure 2: (a) For the 4-orbit toroidal map M = {4,4}a,ß, a = 4 + 3i, ß = -1 + 3i the complex number щ = 6 - 3i generates the unique minimal rotary covering toroidal map {4,4}6-3i, which can be obtained by gluing 3 fundamental regions of the initial map. Meanwhile, the unique minimal regular covering toroidal map for M is {4,4} 15, generated by П2 = 15, can be obtained by gluing 15 fundamental regions. (b) For the 2-orbit toroidal map {4,4}aß, a = 3 + 3i, ß = 2 - 2i the minimal rotary and regular covering toroidal maps coincide with the regular toroidal map {4,4}n, n = 6 - 6i, which can be obtained by gluing together 6 fundamental regions of the initial map. Figure 3: (a) For the 6-orbit toroidal map M = {3, 6}a,ß, а = 3 - 2ш, ß = 1 + 4ш the unique minimal rotary covering toroidal map is {3, 6}ni with ni = 4 + 2ш, which can be obtained by gluing 2 fundamental regions of the initial map. Meanwhile, the unique minimal regular covering map for M is {3, 6}П2 with n2 = 14, obtained by gluing 14 fundamental regions. (b) For the 3-orbit toroidal map {3,6}a^, а = 5 - ш, ß = 4 + ш the minimal rotary and regular covers coincide and are equal to {3, 6}n, n = 6 - 3ш, which can be obtained by gluing 3 fundamental regions. 7 Concluding remarks As noted in Section 1, this paper is part of a larger project which seeks to understand the structure of highly symmetric covers of various maps and polytopes. While many of the results in this area have concerned minimal regular covers, there are other natural questions to consider. For example, if instead we are given a map on the torus which is not equivelar, the minimal regular and rotary covers will not be maps on the torus. Instead, it is natural to try and construct toroidal covers of these maps which have the greatest possible symmetry. Such a question is being considered in a natural successor of this paper (see [5]). Acknowledgments Much of this work was done while the authors were involved in the Fields-MITACS 2011 Summer Undergraduate Research Program. We would like to express our deep gratitude to the Fields Institute for the hospitality and the creative atmosphere. We also would like to thank Daniel Pellicer, Isabel Hubard, and Asia Weiss for their constant support, and for providing the question which has been solved in this paper. The first named author also would like to thank Maksym Skoryk and Alexander Borisenko for fruitful discussions. Both authors thank the anonymous referees for many valuable remarks and improvements. References [1] U. 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Williams, Representing the sporadic Archimedean polyhedra as abstract polytopes, Discrete Math. 310 12 (2010), 1835-1844. [9] M. I. Hartley, D. Pellicer and G. Williams, Minimal covers of the prisms and antiprisms, Discrete Math 312 20 (2012), 3046-3058. [10] I. Hubard, Two-orbit polyhedra from groups, J. Combin. 31 (2010), 943-960. [11] I. Hubard, A. Orbanic, D. Pellicer and A. I. Weiss, Symmetries of equivelar 4-toroids, Discrete Comput. Geom. 48 (2012), 1110-1136. [12] K. Ireland and M. Rosen, A classical introduction to modern number theory, 2nd ed., SpringerVerlag, New-York, 1990. [13] P. McMullen and E. Schulte, Abstract Regular Polytopes, Encyclopedia of Math. Appl., vol. 92, Cambridge University Press, Cambridge, 2002. [14] B. Monson, D. Pellicer and G. Williams, The Tomotope Ars Math. Contemp. 5 (2012), 355370. [15] B. Monson, D. Pellicer and G. Williams, Mixing and Monodromy of Abstract Polytopes. Trans. Amer. Math. Soc. 366 (2014), 2651-2681. [16] S. Negami, Uniqueness and faithfullness of embedding of toroidal graphs, Discrete Math. 44 (1983), 161-180. [17] D. Pellicer and G. Williams, Minimal covers of the Archimedean tilings, Part I, Electron. J. Comb. 19 (2012). [18] J. Siran, Regular maps on a given surface: a survey, in: M. Klazar, J. Kratochvil (eds.), Topics in Discrete Mathematics. Algorithms and Combinatorics, vol. 26, Springer, Berlin, 2006, 591609. [19] J. Tits, A local approach to buildings, in: C. Davis, B. Grünbaum (eds.), The Geometric Vein: the Coxeter Festschrift, Springer-Verlag, NewYork - Heidelberg - Berlin, 1981, 519-547. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 93-107 Group distance magic labeling of direct product of graphs Marcin Anholcer Poznati University of Economics, Faculty of Informatics and Electronic Economy Al. Niepodleglosci 10, 61-875 Poznan, Poland Sylwia Cichacz * AGH University of Science and Technology, Faculty of Applied Mathematics Al. Mickiewicza 30, 30-059 Krakow, Poland Iztok Peterin, Aleksandra Tepeh University of Maribor, Faculty of Electrical Engineering and Computer Science Smetanova 17, 2000 Maribor, Slovenia Institute of Mathematics, Physics and Mechanics Jadranska 19, 1000 Ljubljana, Slovenia Received 16 January 2013, accepted 28 August 2013, published online 11 July 2014 Let G = (V, E) be a graph and Г an Abelian group, both of order n. A group distance magic labeling of G is a bijection I : V ^ Г for which there exists m G Г such that J2xen (v) Kx) = M for all v G V, where N (v) is the neighborhood of v. In this paper we consider group distance magic labelings of direct product of graphs. We show that if G is an r-regular graph of order n and m = 4 or m = 8 and r is even, then the direct product Cm x G is Г-distance magic for every Abelian group of order mn. We also prove that Cm X Cn is Zmn-distance magic if and only if m G {4, 8} or n G {4,8} or m, n = 0 (mod 4). It is also shown that if m, n = 0 (mod 4) then Cm X Cn is not Г -distance magic for any Abelian group Г of order mn. Keywords: Abelian group, distance magic labeling, group labeling, direct product of graphs. Math. Subj. Class.: 05C15, 05C22, 05C25, 05C76 *The author was partially supported by National Science Centre grant nr 2011/01/D/ST1/04104. E-mail addresses: m.anholcer@ue.poznan.pl (Marcin Anholcer), cichacz@agh.edu.pl (Sylwia Cichacz), iztok.peterin@um.si (Iztok Peterin), aleksandra.tepeh@um.si (Aleksandra Tepeh) Abstract 1 Introduction and preliminaries All graphs considered in this paper are simple finite graphs. We use V(G) for the vertex set and E(G) for the edge set of a graph G. The neighborhood N(x) or more precisely NG(x), when needed, of a vertex x is the set of vertices adjacent to x, and the degree d(x) of x is |N(x) |, the size of the neighborhood of x. By Cn we denote a cycle on n vertices. We recall two out of four standard graph products (see [8]). Let G and H be two graphs. Both, the Cartesian product GO H and the direct product G x H are graphs with the vertex set V(G) x V(H). Two vertices (g, h) and (g', h') are adjacent in: • GOH if g = g' and h is adjacent to h' in H, or h = h' and g is adjacent to g' in G; • G x H if g is adjacent to g' in G and h is adjacent to h' in H. Distance magic labeling (also called sigma labeling) of a graph G = (V (G), E (G)) of order n is a bijection i : V ^ {1,...,n} with the property that there is a positive integer k (called magic constant) such that w(x) = J2 yeNa (x) i(y) = k for every x G V (G), where w(x) is the weight of vertex x. If a graph G admits a distance magic labeling, then we say that G is a distance magic graph. See [2] (and also [7]) for the survey on distance magic graphs. The idea of distance magic labeling of graphs has been motivated by the constructions of magic squares. However, finding an r-regular distance magic graph is equivalent to finding equalized incomplete tournament EIT(n, r) [6]. In an equalized incomplete tournament EIT(n, r) of n teams with r rounds, every team plays exactly r other teams and the total strength of the opponents that team i plays is k. Some graphs which are distance magic among (some) products can be seen in [1, 3, 4, 5, 9, 10]. Recently a subclass of distance magic graphs was introduced in [1] that behave nicely among products. A distance magic graph G is called balanced if there exists a bijection i : V (G) ^ {1,..., |V (G)|} such that for every w G V (G) the following holds: if u G N (w) with i(u) = i, then there exists v G N (w), v = u, with i(v) = |V (G)| + 1-i. We say that u is the twin vertex of v (and vice versa) and i is called a balanced distance magic labeling. It is easy to see that a balanced distance magic graph has an even number of vertices and that it is an r-regular graph for an even r. Simple examples are empty graph on an even number of vertices, cycle C4, and K2n - M, for a perfect matching M. The following theorem was proved in [1] and will be used in the second section. Theorem 1.1 ([1]). The direct product G x H is a balanced distance magic graph if and only if one of the graphs is balanced distance magic and the other one is regular. Group distance magic labeling of graphs was recently introduced by Froncek in [5] as in some sense a generalization of distance magic labeling. Let Г be a finite Abelian group of order n. A Г-distance magic labeling of a graph G with |V(G)| = n is an injection from V to Г such that the weight of every vertex x G V is equal to the same element ^ G Г, called the magic constant. If there exists a Г-distance magic labeling of G, we say that G is a Г-distance magic graph. A graph G is called a group distance magic graph if there exists a Г-distance magic labeling for every Abelian group Г of order |V(G)|. The connection between distance magic graphs and Г-distance magic graphs is as follows. Let G be a distance magic graph of order n. If we replace n in {1,... ,n} by 0, we obtain a Zn-distance magic labeling. Hence every distance magic graph is a Zn-distance magic graph. The question remains what happens when we replace Zn by some other Abelian group, and which graphs are Г-distance magic but not distance magic. The following theorem was proved in [5]: Theorem 1.2 ([5]). The Cartesian product CmDCk, m, k > 3, is a Zmk-distance magic graph if and only if km is even. Froncek also showed that the graph C2kdC2k has a (Z2)2k-distance magic labeling for k > 2 and m = (0,0,..., 0) ([5]). It seems that the direct product is the natural choice among (standard) products to deal with Г-distance magic graphs and group distance magic graphs in general. The reason for this is that the direct product is suitable product if we observe graphs as categories. Hence it should perform well with the product of (Abelian) groups. The confirmation of this will be illustrated in the first theorem of each forthcoming section. This fact also makes the direct product the most natural among graph products, but on the other hand the most difficult to handle. Namely, G x H does not need to be connected, even if both factors are. More precisely, G x H is connected if and only if both G and H are connected and at least one of them is non-bipartite [11]. The direct product is commutative, associative, and has attracted a lot of attention in the research community in last 50 years. Probably the biggest challenge (among all products) is the famous Hedetniemi's conjecture: X(G x H)=min{x(G),x(H)}. This conjecture suggests that the chromatic number of the direct product depends only on the properties of one factor and not both. This is not so rare and also in this work we show that it is enough for one factor to be a balanced distance magic graph and then the direct product with any regular graph will result in a group distance magic graph. For more about the direct product and products in general we recommend the book [8]. For V (G) = {xo, xi,..., x|v (G)|-i} and V (H ) = {yo, yi,..., y\y (h)|-i} we use V(G x H) = {vij : i G {0,1,..., |V(G)| - 1},j G {0,1,..., |V(H)| - 1}}. The fundamental theorem of finite Abelian groups states that the finite Abelian group Г of order n can be expressed as the direct sum of cyclic subgroups of prime-power order. This implies that Г = Zp«i x ... x Zpmm, where n = П™ 1 p®* and pi for i G {1,..., m} are not necessarily distinct primes. In particular, if n = 0 (mod 4), then we have the following possibilities: Г = Z2 x Z2 x Zpai x ... x Zp 3. This fact will be used often in what follows. Recall that any group element g G Г of order 2 (i.e., g = 0, 2g = 0) is called an involution, and that a non-trivial finite group has elements of order 2 if and only if the order of the group is even. Moreover every cyclic group of even order has exactly one involution. We will use the notation a0 for the identity element of an Abelian group A. In the next section we present some general results about group distance magic label-ings on the direct products. In the last section we concentrate on direct product of cycles. We will prove also that a graph Cm x Cn is Zmn-distance magic if and only if m G {4,8} or n G {4,8} or m, n = 0 (mod 4). Moreover, we will show that if m, n = 0 (mod 4) then Cm x Cn is not Г-distance magic for any Abelian group Г of order mn. 2 General results We start with the following general theorem for direct product of graphs: Observation 2.1. If an ri-regular graph G1 is ^-distance magic and an r2 -regular graph G2 is Г2-distance magic, then the direct product G1 x G2 is Г1 x Г2-distance magic. Proof. Let ti : V(Gì) ^ Г be a ^-distance magic labeling, and m the magic constant for the graph Gi, i e {1,2}. Define the labeling t : V(G1 x G2) ^ Г1 x Г2 for G1 x G2, as: l((x,y)) = (h (x),t2(y)). Obviously, t is a bijection and moreover, for any (u, w) e V(G1 x G2): w(u,w) = E t(x,y)= I r2 E t1(x),r1 E t2(y) I , (x,y)eNa1xG2 (u,w) \ xeNGl (u) yENa2 (w) J w(u,w) = (Г2Р1,Г1 P2) = M, which settles the proof. □ Theorem 2.2. If G is a balanced distance magic graph, then G is a group distance magic graph. Proof. Let G be a balanced distance magic graph of order n. Recall that n is an even number and G is an r-regular graph for an even r. For any Abelian group Г of order n holds Г = Z2t xA for some natural number t > 0 and some Abelian group A of order f .If g e Г, then we can write g = (j, ai) where j e Z2t and ai e A for i e {0,1,..., f - 1}. Let V (G) = {u1, u2,..., u n, u1, u2,..., u'n }. For i e {1,..., f} we define the following labeling t for a vertex ui and its twin vertex ui: t(ui) = ((i - 1)(modt),ayi-1^ and t(ui) = (2t - 1,a0) - t(ui). Since t(ui) + t(ui) = (2t - 1, a0) for every i e {1,..., f }, we get w(v) = E t(u) = E (t(ui)+ t(ui)) = ueN (v) UiEN(v) = E (2t - 1,ao) = 2(2t - 1,ao), UiEN (v) where § is an integer since r is even. Moreover, every element of Г is used exactly once and so G is Г-distance magic. □ Since for any graph G of order m, the graph Kn x G is isomorphic to Knm, by Theorems 1.1 and 2.2 the next result immediately follows. Theorem 2.3. If G is a balanced distance magic graph and H an r-regular graph for r > 1, then G x H is a group distance magic graph. □ Notice that by the above Theorem 2.3, if G is an r-regular graph (r > 1), then the graph C4 x G is a group distance magic graph. We cannot generalize this result to other cycles than C4. Namely, Cn x K2 is isomorphic to 2Cn (i.e., the union of two cycles Cn) when n is even and to C2n when n is odd. It is easy to see that both 2Cn for n = 4 and C2n for n = 2 are not Г- distance magic for any Abelian group Г of order 2n (for n > 5, in both cases under the assumption that there is some group distance magic labeling t, we obtain t(vi) = t(vi+4) for all the vertices vj, and we easily derive a contradiction also for n = 3). Nevertheless, for many regular graphs the result still holds. For C8 as we will see next. Theorem 2.4. If G is an r-regular graph of order n for some even r, then direct product C8 X G is a group distance magic graph. Proof. Let V(G) = {xo, x\,..., xn-i} be the vertex set of G, let C8 = uoui... u7u0, and H = C8 X G. Notice that if xpxq e E (G), then vj,q e NH (vip) if and only if j e {i — 1 , i + 1} (where the sum on the first suffix is taken modulo 8). We are going to consider three cases, depending on the structure of Г. Case 1: Г = Z2 x Z2 xA for some Abelian group of order 2n. We can write g e Г as (j1,j2, ak) for j1,j2 e Z2 and ak e A for k e {0,1,..., 2n — 1}. For j e {0,1,..., n — 1} we set ( (0,0,a2j+i), if i e {0,1}, t(vij) = { (0,1,a2j+i-4), if i e {4, 5}, [ (1,1,ao) — t(vi-2,j), if i e {2, 3, 6, 7}. Clearly, t : V(C8 x G) ^ Г is a bijection and t(viij-) + t(vi+2j) = (yi,a0), where yi e {(1,1), (1,0)}, and so 2yi = (0,0). Hence for every i e {0,1,..., 7} and j e {0, 1, . . . , n — 1} we get w(vi,j ) = E (t(vi-i,p) + t(vi+i,p)) = E (yi-i,ao) = xpeNa(xj ) xpeNa(xj ) r = 2(0, 0, ao) = (0, 0, ao) and C8 X G is Г-distance magic since r is even. Case 2: Г = Z4 xA for some Abelian group A of order 2n. If g e Г, then we can write g = (j, ak ) for j e Z4 and ak e A for k e {0,1,..., 2n — 1 }. For j e { 0, 1 , . . . , n — 1 } we define ( (0,a2j+i), if i e{0, 1}, t(vi,j)= < (2,a2j+i-4), if i e {4, 5}, [ (3, ao) — t(vi-2,j), if i e {2, 3, 6, 7}. Again t : V (C8 x G) ^ Г is obviously a bijection and t(viij- )+t(vi+2jj ) = (yi, ao), where yi e {1, 3}, and thus 2yi = 2. Hence for every i e {0,1,..., 7} and j e {0,1,..., n — 1} we get w(vi,j )= E (t(vi-i,p )+ t(vi+i,p)) = E (yi-i,ao) = 2 (2,ao) = (r,ao) xp£NG(xj ) xpENG (xj ) and C8 x G is Г-distance magic. Case 3: Г = Z2a xA for a > 2 and some Abelian group A of order ^О-з. If g G Г, we can write g = (p, ak ) for p G Z2a and ak G A for k G {0,1,..., 1}. For j g {0,1,..., 225=3 — 1} define the following labeling t ( ((2j + i)(mod2a-2), a j-2-а+зj) , if i G {0,1}, t(vi,j ) = I (2a-1, ao) + t(vi_4,j) , if i G {4, 5}, [ (2a — 1,ao) — t(vi-2j ), if i G {2, 3, 6, 7}. As in previous cases t : V(C8 x G) ^ Г is a bijection and t(vi,j) + t(vi+2,j) = (yi, a0) for some yi G {2a-1 — 1,2a — 1}. Thus 2(t(vi,j) + t(vi+2,j)) = (2yi, ao) = (—2, ao). For every i G {0,1,..., 7} and j G {0,1,..., n — 1} we get w(vi,j) = (t(vi-i,p) + t(vi+i,p)) = (yi-1, ao) = xpeNa(xj ) xpeNa(xj ) r = 2, ao) = (—r, ao) and C8 X G is Z2a xA-distance magic. □ The natural question arises whether we can prove similar results for every cycle Cn where n = 0 (mod 4). The answer to this question is negative as we will see in the next section. It will also be clear from the following section why we cannot expect similar results for n = 0 (mod4). (For both claims see Theorem 3.5.) However, below we give some groups Г such that for G being an r-regular graph of order n for some even r, the direct product C2p x G admits a Г-distance magic labeling. Proposition 2.5. If G is an r-regular graph of order n for some even r, then the direct product C2p X G, p > 2, admits an Ax B-distance magic labeling for any Abelian group B of order n and an Abelian group A such that: • A = (Z2)p, • A = Z4 x(Z2)p-2, • A = Z8 x(Z2)p-3, • A= (Z4)2 x (Z2)p-4. Proof. Let V(G) = {xo, x1,..., xn-1} be the vertex set of G, let C2p = uou1... u2p-1uo, and H = C2p x G. Notice that if xpxq G E(G), then vj,q G NH(vi,p) if and only if j G {i — 1, i + 1} (where the sum on the first suffix is taken modulo 2p). Let the elements of B be bo, b1,..., bn-1. Recall that for any element r G (Z2)p we have 2r = 0. Case 1: A = (Z2)p. Let the elements of (Z2)p be ro,..., r2p-1. Each element of Г = (Z2)p x B can be thus expressed as (ri, bj), where ri G (Z2)p and bj g B. We define the labeling t as follows: Г (ri, bj), if i(mod4) G{0,1}, t(vi,j) i (ri, —bj), if i(mod4) G {2, 3}. It is straightforward to check that i is bijective and £(vi,j )+ i(vi+2,j ) = (r + ri+2, bo). Hence for every i G {0,1,..., 2p - 1} and j G {0,1,... ,n — 1} we get w(vi,j) = Y (i(vi-i,d) + i(vi+i,d)) = Y (ri-i + ri+l, bo) = XdeNa(xj) XdeNa(xj ) r = 2(ro,bo) = (ro,bo) and C2p x G is (Z2)p x B-distance magic since r is even. Case2: A = Z4 x(Z2)p-2. Let the elements of (Z2)p-2 be ro,..., r2p-2-1. Each element of Г = Z4 X (Z2)p-2 X B can be thus expressed as (q, ri, bj ), where q G Z4, ri G (Z2)p-2, and bj g B. We define the labeling i in the following way i(v ) f (q,ri,bj), if q G {0, 1} i(v4i+q,j) I (q, ri, —bj), if q G {2, 3}. Again it is easy to check that i is bijective and i(vi,j) + i(vi+2,j) = (yi, zi, bo), where yi G {0,2} and zi = rLi/4j + rL(i+2)/4j, and so 2(y>,zi) = (0,ro). Hence for every i G {0,1,..., 2p — 1} and j G {0,1,..., n — 1} we get w(vi,j) = Y (i(vi-i,d)+ i(vi+i,d)) = Y (yi-i, zi-i, bo) = xdeNa(xj ) xdeNa(j ) r = 2 (0, ro, bo) = (0, ro, bo) and C2p X G is Z4 x (Z2)p-2 x B-distance magic since r is even. Case 3: A = Z8 x(Z2)p-3. Let the elements of (Z2)p-3 be ro,..., r2p-3-i. Each element of Г = Z8 x(Z2)p-3 x B can be thus expressed as (a(q),ri,bj-), where q G Z8, ri G (Z2)p-3, bj G B, and the function a : Z8 ^ Z8 is defined as 2, admits an Ax B-distance magic labeling for any Abelian group B of order n and an Abelian group A such that: • A = Z8 X Z4 X (Z2 )p—4, • A = Zi6 X (Z2 )p—3. Proof. Let V(G) = {x0, xi,..., xn-i} be the vertex set of G, let C2p = u0ui... u2p — iu0, and H = C2p X G. Notice that if xpxq G E(G), then vjq G (vijP) if and only if j G {i - 1, i + 1} (where the sum on the first suffix is taken modulo 2p). Let the elements of B be b0, bi,..., bn/2-i. Case 1: A = Z8 x Z4 x(Z2)p-4. Each element of Г = Z8 x Z4 x(Z2)p-4 x B can be expressed as (a(q),ri,bj-), where q G Z32, ri G (Z2)p-4, bj G B, and the function o : Z32 ^ Z8 X Z4 is defined as: i 0 1 2 3 4 5 6 7 o(i) (0,0) (1,0) (3,1) (2,1) (4,2) (5,2) (7,3) (6,3) i 8 9 10 11 12 13 14 15 o(i) (0,2) (1,2) (3,3) (2,3) (4,0) (5,0) (7,1) (6,1) i 16 17 18 19 20 21 22 23 o(i) (0,3) (1,3) (3,2) (2,2) (4,1) (5,1) (7,0) (6,0) i 24 25 26 27 28 29 30 31 o(i) (0,1) (1,1) (3,0) (2,0) (4,3) (5,3) (7,2) (6,2) For j = 0 (mod 2) we define the labeling t in the following way: t(vi6i+t,j ) = (o(t),ri,bLjj), if t(mod4) G {0,1}, (o(t),ri, -bL jj), if t(mod4) G {2, 3}; and for j = 1 (mod 2) we set i(vi6 i+t,j ) = (a(16 + t),Vi,by j j), if t(mod4) G {0,1}, (a(16 + t),Vi, -bjjj), if t(mod4) G {2, 3}. It is straightforward to check that i is bijective and £(vitj) + i(vi+2j-) = (yi, zi, b0), where yi G {(3,1), (7, 3), (7,1)} and z = rLi/i6j + rL(i+2)/i6j+i, so 2(yi,Zi) = (6, 2,ro). Hence for every i G {0,1,..., 2p — 1} and j G {0,1,..., n — 1} we get w(vi,j )= £ (£(vi-i,d)+ ^(vi+i,d)) = £ (yi-i, zi-i, bo) = 2 (6, 2,ro,bo) Xd&Na(xj ) xdeNa(xj ) and C2p X G is Z8 x Z4 x (Z2 )p-4 x B-distance magic since r is even. Case 2: A = Zi6 x(Z2)p-3. Each element of Г = Zi6 x(Z2)p-3 xB as (a(q),n ,bj ), where q G Zi6, ri G (Z2)p-3, bj G B, and the function a : Zi6 ^ Zi6 is defined as: i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 a(i) 0 2 1 15 8 10 9 7 4 6 13 11 12 14 5 3 We define the labeling i for j = 0 (mod 2) as ' (a(t),ri,bLjj), if t(mod4) G {0,1}, i(v8i+t,j ) = and for j = 1 (mod 2) by (a(t),ri, -bL jj), if t(mod4) G {2, 3}, (a(8 + t),ri,bLjj), if t(mod4) G {0,1}, (a(8 + t),ri, —ž? L j j), if t(mod4) G {2, 3}. i(v8i+t,j --------- L 2 J Again it is easy to see that i is bijective and i(vi,j) + i(vi+2,j) = (yi, zi,b0), where yi G {1,9} and zi = rLi/8j + rL(i+2)/8j, so 2(yi,zi) = (2,ro). Hence for every i G {0,1,..., 2p — 1} and j G {0,1,..., n — 1} we get w(vi,j)= £ (i(vi-i,d) + i(vi+M))= 53 (yi-i, Zi-i, bo) = ^(2, ro,bo) Xd&Na(xj ) XdENa(xj) and C2p x G is Zi6 x (Z2)p 3 x B-distance magic since r is even. □ 3 Г-distance magic labeling of Cm x Cn In this section we concentrate on the direct product of two cycles. Theorem 3.1. If m, n = 0 (mod 4), then the direct product Cm x Cn is Ax B-distance magic for any Abelian groups A and B of order m and n, respectively. Proof. Let Г = A x B and let b0 be the identity of B. We consider three cases, depending on the factorization of Г. Case 1: Г = Z2 x Z2 xGxß for some Abelian group G of order m. If g G Г, then we can write g = (j\,j2,ak, b j) for jl,j2 G Z2, ak G G for k G {0,1,..., m - 1}, and bj G B for j G {0,1,..., n - 1}. For i G {0,1,..., m - 1} we set: é(v ) = j {0,l,aìлi,Ъj), if i(mod4) G {0,1}, (1,1,ao,bo) - i(vi-2,j ), if i (mod 4) G {2, 3}, for j(mod4) G {0, 1} and ) = j (0,i, -aLiJ,bj), if i(mod4) G{0, 1} ( i,j ) 1(1,1,ao,bo) - t(vi-2,j ), if i (mod 4) G {2, 3}, for j(mod4) G {2, 3}. It is easy to see that £ : V(Cm x Cn) ^ Г is a bijection and w(vitj) = £(vi-1,j-1) + £(vi_i,j+i) + ^(vi+i,j_i) + ^(vi+i,j+i) = (0, 0, a o, bo). Case 2: Г = Z4 xG x B for some Abelian group G of order m. If g G Г, then we can write g = (q, ak, bj ) for q G Z4, ak G G for k G {0,1,..., m -1}, and bj G B for j G {0,1,..., n - 1}. For i G {0,..., m - 1} we define the following labeling £: ) = { for j (mod 4) G {0,1} and (i, aL i j, b j ), if i(mod4) G {0,1}, (3,aoA) - £(vi_ 2,j ), if i(mod4) G {2, 3}, ) I (i, -aLjj,bj), if i(mod4) G{0,1}, ( ) \ (3,ao,bo) - £(vi_2,j), if i(mod4) G {2, 3}, for j(mod4) G {2, 3}. Again £ : V(Cm x Cn) ^ Г is a bijection and w(vž,j) = £(vi_i,j_i) + £(vi_i,j+i) + £(vi+i,j_i) + £(vi+i,j+i) = (2, ao, bo). Case 3: Г = Z2a xG x B for a > 2 and some Abelian group G of order m. Notice that this case is meaningful only when m = 0 (mod 8). If g G Г, then we can write that g = (q, ak, bj) for q G Z2a, ak G G for k G {0,1,..., 2m - 1} and bj G B for j G {0,1,..., n - 1}. For i G {0,..., m - 1} we define the following labeling £: ((i(mod2a))2a_2, aLi/2aj,bj) , if i(mod2a) G {0,1}, 2a))2a_2, £(vi,j )={ ((1 - i(mod2a))2a_2, -aLi/2aj, -b,) , if i(mod2a) G {2, 3}, (1,ao,bo) + £(vi_4,j ), if i (mod 2a) G {0,1, 2, 3}, for j(mod4) G {0, 1} and ( (-(i(mod 2a))2a_2 - 1, -aLi/2aJ, -bj-) , if i(mod2a) G {0,1}, £(vi,j)=< ((i(mod2a - 1))2a_2 - 1,aLi/2aj,bj) , if i(mod2a) G {2, 3}, ( (-1,ao,bo) + £(vi_4,j), if i (mod 2a) G {0,1, 2, 3}, for j(mod4) G {2, 3}. It is easy to see that £ : V(Cm x Cn) ^ Г is a bijection and moreover, w(vi,j) = £(vi_i,j-i) + £(vi+iij_i) + £(vi_iij+i) + £(vi+iij+i) = (—2, ao, bo). □ One can ask if it is possible to find an Ax B -distance magic labeling of Cm x Cn, if |A| > m. A partial answer is given by the following observation. Proposition 3.2. If m, n = 0 (mod 4), then the direct product Cm x Cn is Zt xA -distance magic for m|t and any Abelian group A order mn. Proof. Let Г = Zt xA where m|t and A is an Abelian group of order mn. If g g г, then we can write that g = (j, ak ) for j g Zt and ak gA for k g {0,1,..., mn — 1}. For i g {0,1,..., m — 1} let jm(mod t) 2 , jm(mod t) t L jm J £(vi,j ) for j (mod 4) g {0,1} and £(vi,j ) = < 2 1 4 jm(mod t) l jm j -m ,-a 4 L T J, jm(mod t) 2 2 , aLjpJ £(vi-4,j) + (1, ao), jm(mod t) 1 a 2 1, aL jm J jm(mod t) m 2 T — 1, — aLjm J jm(mod t) m -t--1 л--1, jm(mod t) i m -, 2 + 2 1, aL^rJ, L jm j, £(vi-4,j ) + ( — 1,ao), if i = 0 if i = 1 if i = 2 if i = 3 if i > 3, if i = 0 , if i = 1 if i = 2 if i = 3 if i > 3, for j (mod 4) g {2, 3}. Notice that we obtain mn/t blocks such that in every block we have all elements from Zt as the first coordinate. Moreover for i g {0,1,..., mn/t — 1} in i-th block we have labels (j, ai), where j g {0,1,..., t/2 — 1}. Therefore £ is bijective and p = (—2, a0) is the magic constant. □ The above results encourage us to post the following conjecture. Conjecture 3.3. If m, n = 0 (mod 4), then Cm x Cn is a group distance magic graph. Now we are going to present some sufficient conditions for a graph G not to be group distance magic. Theorem 3.4. Assume that m, n > 3, m, n g {4,8}, m = 4b + d and n = 4a + c for some integers a, b > 0 where c g {0,1, 2, 3} and d g {1,2, 3}. If an Abelian group Г of order mn has less than max{2, a — 1} involutions, then Cm x Cn is not Г-distance magic. Proof. Let m, n, a, b, c, d be as in the statement of the theorem. Thus m = 0 (mod 4). Let G = Cm x Cn. Assume that there exists a group Г of order mn such that G is Г-distance magic, i.e., there is a bijection £ : V (G) ^ Г such that for every x g V (G), w(x) = p for some constant p g Г. Furthermore, let g0 be the identity of Г. 2 For any integers i, p, and s we have w(vi+p+ 1,s+l) = £(vi+p,s) + £(vi+p,s+2) + £(vi+p+2,s) + £(vi+p+2,s+2) = M, where the first suffix is taken modulo m, and the second one modulo n. Comparing the above equality for p = 0 and p = 2, we obtain ć(vi,s ) + ^(vi,s+2) = ^(vi+4,s) + ^(vi+4,s+2). More generally, if we consider the equality for p = j and p = j + 2 for some integer j, we obtain that Z(vi+j,s) + ć(vi+j,s+2) = £(vi+j+4,s) + ^(vi+j+4,s+2) for every j. In consequence, ^(vi,s) + ^(vi,s+2) = ^(vi+4j,s ) + ^(vi+4j,s+2) (3.1) for every j .As m = 0 (mod 4), there exists such a j that i + 4 j = i + 2 (mod m). This way we obtain from (3.1) £(vi,s ) + ^(vi,s+2) = £(vi+2,s) + ^(vi+2,s+2). (3.2) Substituting s with s + 2 in (3.2) we obtain 2) + ^(vi,s+4) = ^(vi+2,s+2) + ^(vi+2,s+4) (3.3) and finally by subtracting (3.3) from (3.2) £(vi,s ) - ^(vi,s+4) = ^(vi+2,s) - ^(vi+2,s+4). (3.4) In a similar way as (3.1) we can prove that for any i, j, and s ^K,s) + ^K+2,s) = ^(vi,s+4j ) + ^(vi+2,s+4j ). (3.5) In particular from (3.5) for j = 1 we get ^(vi,s) - ^(vi,s+4) = ^(vi+2,s+4) - ^(vi+2,s). (3.6) This leads, if we add together (3.4) and (3.6), to 2(^(vi,s) - ^(vi,s+4)) = go. By comparing the last equality for s = 4p and s = 4(p + 1), where p is any nonnegative integer, we can observe that 2(^(vi,o) - ^(vi,4p)) = go at least for every 1 < p < a - 1. This bound is sharp when c = 0. Let first a > 2. We have at least 2№i,o) - ^(vM)) = go and 2(^(vi,o) - ^Ks)) = go. (3.7) On the other hand, if a < 2, then c = 0, since n {4,8}. Hence vi,o, vi,4, and vi,8 are again different vertices and (3.7) holds as well. Fix i = 0. Since t is bijection, t(v0,0) - t(v0,4p) = 0 for every p such that p G {1,..., a — 1}, therefore t(v0,0)—t(v0,4p) has to be an involution for every p G {1,..., a — 1} and we have at least two involutions when n = 3. Moreover, bijectivity of t implies that t(v0,0) — t(v0,4pi ) = t(v0,0) — t(v0,4p2 ) for every pi = p2 such that pi,p2 G {1,..., a — 1}. Thus the set of involutions of Г has to consists of at least max{2, a — 1} distinct elements. □ Theorem 3.5. If m, n = 0 (mod 4) then Cm x Cn is not Г-distance magic for any Abelian group Г of order mn. Proof. If m, n are odd, then any group Г has odd order mn and we are done by Theorem 3.4, as there are no involutions in Г. If m = 2 (mod 4) and n is odd (or n = 2 (mod 4) and m is odd, resp.), then mn = 2 (mod 4). Thus by the fundamental theorem of finite Abelian groups Г = Z2 x Z„«i x ... x Z„«fc where mn = 2 Пi=1 p^ and pj > 2 p 1 pk for i g {1,..., k} are not necessarily distinct primes. Therefore there exists exactly one involution i in Г (i = (1,0,..., 0)) and we are done by Theorem 3.4. Suppose now that m, n = 2 (mod 4), then m = 2 + 4a, n = 2 + 4b and mn = 4(1 + 2a)(1 + 2b) for some integers a, b. Thus by the fundamental theorem of finite Abelian groups Г = Z2 x Z2 x A or Г = Z4 x A for some Abelian group A of order (1 + 2a)(1 + 2b). Since (1 + 2a)(1 + 2b) is an odd number, then, if Г = Z4 x A, there exists only one involution i = (2,0) in Г and Cm x Cn is not Z4 x A-distance magic by Theorem 3.4. In the case Г = Z2 x Z2 x A, there exist exactly three involutions i1 = (1,0,0), i2 = (0,1,0) and i3 = (1,1,0) in Г. Let G = Cm x Cn and assume that G is Г-distance magic, i.e., there is a bijection t : V (G) ^ Г such that for every x g V (G), w(x) = m for some constant m. Using the same arguments as in the proof of Theorem 3.4, since m = 2 (mod4), we obtain that 2(t(v0,0) + t(v0,2)) = 2(t(v0,2) + t(v0,4)) = M and 2(t(v0,4) + t(v0,6)) = 2(t(v0,6) + t(v0,8)) = M. On the other hand since n = 2 (mod4) we have gcd(2, n) = gcd(4, n) = 2 and there exists a' such that 4a' = 2 (modn). By repeating the above arguments we get 2(t(v0,4) + t(v2,4)) = 2(t(v2,4) + t(v4,4)) = m and 2(t(v0,4) + t(vm_2,4)) = 2(t(vm_2,4) + t(vm_4,4)) = m. Therefore: 2(t(v0,0) — t(v0,4)) = 2gi = 0, 2(t(v0,8) — t(v0,4)) = 2g2 = 0, 2(t(v4,4) — t(v0,4)) = 2g3 = 0 2(t(vm_4,4) — t(v0,4))=2g4 = 0. If any g =0 (for i G {1, 2,3,4}), then the labeling t is not a bijection as m, n = 0 (mod 4). Thus we can assume that all g are involutions and by the Pigeonhole Principle there exist j = i such that g = g (since there are only three involutions in Г) what implies that the labeling t is not a bijection m, n = 0 (mod 4) (e.g., if g2 = g3, then t(v0,8) = t(v0,4)), a contradiction. □ The immediate corollary follows. Corollary 3.6. Assume that m, n > 3 and {m, n} = {4a, 4b + c} for some integers a > 3 and b > 0, c G {1, 2, 3}. Then Cm x Cn can be Г-distance magic only in the following cases: • c e {1, 3} and r = Ax (Z2)p+2 for some Abelian group A of odd order, where a = 2p, • c e {1, 3} and Г = Ax Z3 x (Z2)p for some Abelian group A of odd order, where a = 3 ■ 2p-2, • c e {1,3} and Г = A x (Z2)p x Z4 for some Abelian group A of odd order, where a = 2p, • c e {1, 3} and Г = Ax (Z2)p-2 x (Z4)2 for some Abelian group A of odd order, where a = 2p, • c e {1,3} and Г = Ax (Z2)p-1 x Z8 for some Abelian group A of odd order, where a = 2p, • c =2 and Г = A x (Z2 )p+3 for some Abelian group A of odd order, where a = 2p, • c = 2 and Г = Ax Z3 x (Z2)p+1 for some Abelian group A of odd order, where a = 3 ■ 2p-2, • c = 2 and Г = A x Z3 x(Z2)p-1 x Z4 for some Abelian group A of odd order, where a = 3 ■ 2p-2, • c = 2 and Г = Ax Z5 x (Z2 )p for some Abelian group A of odd order, where a = 5 ■ 2p-3, • c = 2 and Г = Ax Z7 x (Z2 )p for some Abelian group A of odd order, where a = 7 ■ 2p-3, • c = 2 and Г = Ax (Z2)p+1 x Z4 for some Abelian group A of odd order, where a = 2p, • c = 2 and Г = A x (Z2)p-1 x (Z4)2 for some Abelian group A of odd order, where a = 2p, • c = 2 and Г = A x (Z2)p x Z8 for some Abelian group A of odd order, where a = 2p, • c = 2 and Г = Ax (Z2)p-2 x Z4 x Z8 for some Abelian group A of odd order, where a = 2p, • c = 2 and Г = Ax (Z2)p-1 x Z16 for some Abelian group A of odd order, where a = 2p. Proof. Let a = a2p for some odd number a. Observe that the number of involutions is equal to 2e - 1, where ß is the number of the factors Z2k of Г. By Theorem 3.4 we have a2p - 1 < 2e - 1 and hence a < 2e-p. It is straightforward to see that if c e {1,3}, then the maximum number of such factors is p + 2 and a < 4, while in the case when c =2 it is p + 3 and a < 8. Moreover Z6 = Z3 x Z2, Z10 = Z5 x Z2, Z12 = Z4 x Z3 and Z14 = Z7 x Z2 are the only groups of the respective order, so the listed groups are the only ones that consist of at least p factors Z2 k. □ In the previous section in Propositions 2.5 and 2.6 we presented constructions for all the cases from Corollary 3.6, where m = 2p or n = 2p for some integer p. However we think that whole Corollary 3.6 gives not only necessary but also sufficient conditions for a graph to be group distance magic so we post the following conjecture. Conjecture 3.7. Assume that m, n > 3 and {m, n} = {4a, 4b + c} for some integers a > 3 and b > 0, c G {1,2,3}. Then Cm x Cn is Г-distance magic in the following cases: • c G {1,3} and Г = Ax Z3 x (Z2)p for some Abelian group A of odd order, where a = 3 ■ 2p-2, • c = 2 and Г = Ax Z3 x (Z2)p+1 for some Abelian group A of odd order, where a = 3 ■ 2p-2, • c = 2 and Г = A x Z3 x(Z2 )p-1 x Z4 for some Abelian group A of odd order, where a = 3 ■ 2p-2, • c = 2 and Г = Ax Z5 x(Z2)p for some Abelian group A of odd order, where a = 5 ■ 2p-3, • c = 2 and Г = Ax Z7 x(Z2)p for some Abelian group A of odd order, where a = 7 ■ 2p-3, We finish with the following result. Theorem 3.8. A graph Cm x Cn is Zmn-distance magic if and only if m G {4,8} or n G {4, 8} or m,n = 0 (mod4). Proof. If m = 0 (mod 4) and n G {4, 8}, or n = 0 (mod 4) and m G {4,8}, then the group Zmn has at most one involution i (namely i = mp, if mn is even) and so Cm x Cn is not Zmn-distance magic by Theorem 3.4. If n = 4 or m = 4 then Cm x Cn is Zmn-distance magic by Theorem 2.3 and if m, n = 0(mod4), then Cm x Cn is Zmn-distance magic by Proposition 3.2. If n = 8 or m = 8, then the graph Cm x Cn is Zmn-distance magic by Theorem 2.4. □ References [1] M. Anholcer, S. Cichacz, I. Peterin, A. Tepeh, Distance magic labeling and two products of graphs, Graphs Combin., accepted. [2] S. Arumugam, D. Froncek and N. Kamatchi, Distance Magic Graphs - A Survey, Journal of the Indonesian Mathematical Society, Special Edition (2011) 11-26. [3] S. Beena, On E and E labelled graphs, Discrete Math. 309 (2009), 1783-1787. [4] S. Cichacz, Note on group distance magic graphs G[C4], Graphs Combin. 30 (2014), 565-571. [5] D. Froncek, Group distance magic labeling of Cartesian product of cycles, Australasian J. Combin. 55 (2013), 167-174. [6] D. Froncek, P. KoväT and T. KoväTova, Fair incomplete tournaments, Bull. of ICA 48 (2006) 31-33. [7] J. A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., DS6, http://www. combinatorics.org/issue/view/Surveys. [8] R. Hammack, W. Imrich, S. KlavZar, Handbook of Product Graphs, Second Edition, CRC Press, Boca Raton, FL, 2011. [9] M. Miller, C. Rodger and R. Simanjuntak, Distance magic labelings of graphs, Australasian J. Combin. 28 (2003), 305-315. [10] S. B. Rao, T. Singh and V. Parameswaran, Some sigma labelled graphs I, in: S. Arumugam, B.D. Acharya and S.B. Rao (eds.), Graphs, Combinatorics, Algorithms and Applications, Narosa Publishing House, New Delhi (2004) 125-133. [11] P. M. Weichsel, The Kronecker product of graphs, Proc. Amer. Math. Soc. 13 (1962), 47-52. /^creative , ars mathematica ^commons contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 109-113 On spectral radius and energy of complete multipartite graphs Dragan StevanoviC * University of Primorska, UP IAM, Muzejski trg 2, SI-6000 Koper, Slovenia and Mathematical Institute, Serbian Academy of Science and Arts, Knez Mihajlova 36, 11000 Belgrade, Serbia Ivan Gutman Faculty of Science, University of Kragujevac, P.O. Box 60, 34000 Kragujevac, Serbia Masood U. Rehman Air University, Islamabad, Pakistan Received 18 June 2013, accepted 8 March 2014, published online 8 August 2014 Abstract Let Kni ,n2,...,np denote the completep-partite graph, p > 1,on n = n1 + n2 + • • • + np vertices and let n1 > n2 > • • • > np > 0. We show that for a fixed value of n, both the spectral radius and the energy of complete p-partite graphs are minimal for complete split graph CS(n,p — 1) and are maximal for Turan graph T(n, p). Keywords: Spectral radius of graph, graph energy, complete multipartite graph, complete split graph, Turän graph. Math. Subj. Class.: 05C50 1 Introduction Let G be a simple graph on n vertices, and A1 > A2 > • • • > An be its eigenvalues (i.e., the eigenvalues of the (0,1)-adjacency matrix of G) [2, 4]. Then A1 = A1(G) is said to be the spectral radius of the graph G whereas n E = E (G) = Y |Ai| (1.1) i= 1 * Corresponding author. The work of DS was supported by Research Programme P1-0285 and Research Project J1-4021 of the Slovenian Research Agency and Research Grant ON174033 of the Ministry of Education and Science of Serbia. E-mail addresses: dragan.stevanovic@upr.si (Dragan Stevanovic), gutman@kg.ac.rs (Ivan Gutman), (Masood U. Rehman) is its energy. The spectral radius is one of the most thoroughly investigated graph-spectral parameters [2, 4, 3]. Also the graph energy has recently attracted much attention [6, 7]. In spite of this, not much is known on A1 and E of complete multipartite graphs [5]. Let p > 1. Denote by KU1 ,U2,...,Up the complete p-partite graph on n = n1 + n2 + • • • + np vertices. For convenience, we assume that n1 > n2 > • • • > np > 0. Two particular types of complete multipartite graphs are: • Complete split graph CS(n,p - 1) = KU-p+1,1,...,1 consisting of an independent set of n - p +1 vertices and a clique of p - 1 vertices, such that each vertex of the independent set is adjacent to each vertex of the clique, and • Turan graph T(n,p) = K[n/p],...,[u/p],[u/pi,...,[u/pi, the (p + 1)-clique-free graph with maximum number of edges [10]. A fundamental result on the spectrum of complete multipartite graphs is: Theorem 1.1 ([9, 8]). A connected graph has exactly one positive eigenvalue of its adjacency matrix if and only if it is a complete multipartite graph. An immediate consequence of Theorem 1.1 is Lemma 1.2. If A1 is the spectral radius of the complete multipartite graph KUl,U2,...,Up , then E (Kn i,U2,...,up ) = 2 A1. Proof. Since all graph eigenvalues are real numbers and their sum is zero, from Eq. (1.1) follows that E (G) is equal to twice the sum of positive eigenvalues. □ It is known that the characteristic polynomial of KUl,U2,...,Up is given by [2, 5] ф^щ^.^пр, A) = AU-W 1 - ]T J++- J Д (A + n) . (1.2) V i=1 / j=1 The spectrum of KUl,U2,...,Up consists of the spectral radius A1 determined from the equation J2P=1 x+U = 1, eigenvalue 0 with multiplicity n - p and p -1 eigenvalues situated in the intervals [-np, -np-1],..., [-n2, -n1]. In the special case n1 = n2 = • • • = np = t, the spectrum of Kt,t,...,t consists of the spectral radius t(p - 1) with unit multiplicity, eigenvalue 0 with multiplicity p(t - 1), and eigenvalue -t with multiplicity p - 1, so that E(Kt,t,...,t) = 2(p - 1)t. Remark 1.3. The 1-partite complete graph (when p = 1 and t = n) is the edgeless graph KU for which, consistently, A1 = E = 0. The n-partite complete graph (when p = n and t = 1) is the ordinary complete graph KU for which, consistently, A1 = n - 1 and E = 2(n - 1). If p = 2, then the special case of Eq. (1.2) is ф^щ,^ ,A) = AU-2(A2 - n1 n2) , from which the (well known) expressions for spectral radius and energy follow: A1 (Kn1 ,U2 ) = V-1 n2, E (Kn ,n2 ) = 2^n1 n2 . Because n1 + n2 = n, for a fixed number of vertices n, we arrive at: Claim 1.4. 1° Ai (Kni ,n2 ) and E(Kni ,n2 ) are minimal if ni = n — 1 and n2 = 1. 2° Ai(Kni,n2 ) and E(Kni,n2 ) are maximal if ni — n2 < 1. If p = 3, then the special case of Eq. (1.2) is 4>(Ktut2,t3, A) = Ap-3 (A3 — (ti t2 +1213 +13 ti) A — 2ti t213) . (1.3) Using (1.3), it is relatively easy to show the following: Claim 1.5. 1° Let ni + n2 + n3 be equal to a fixed integer n. Then the spectral radius and the energy of Kni,n2,n3 are minimal if ni = n — 2 and n2 = n3 = 1. 2° Let ni + n2 + n3 be equal to a fixed integer n. Then the spectral radius and the energy of Kni,n2,n3 are maximal if ni — n3 < 1. The above claims were the motivation for establishing our main results: Theorem 1.6. Let p > 2 and ni + n2 + ■ ■ ■ + np be equal to a fixed integer n. Then the spectral radius and the energy of Kni,n2,...,np are minimal if Kni,n2,...,np = CS(n,p — 1). Theorem 1.7. Let p > 2 and ni + n2 + ■ ■ ■ + np be equal to a fixed integer n. Then the spectral radius and the energy of Kni,n2,...,np are maximal if Kni,n2,...,np = T(n,p). 2 Proofs of Theorems 1.6 and 1.7 Let Ai and x be, respectively, the spectral radius and the corresponding unit eigenvector of the adjacency matrix of Kni,...,np . Since Ai is a simple eigenvalue of Kni,...,np , similar vertices have equal x-components. Hence, we may denote by xi the common x-component of vertices in the part of Kn i ,...,np having ni vertices for i = 1,..., p. From the eigenvalue equation, we have: p Ai xi = ^2 nk xk = X — ni xi k=i k=i where X = Y7k=i nk xk . Then X (2.1) Ai + ni Lemma 2.1. If ni — nj > 2, then Ai(Kni,...,ni-i,...,nj + i,...,np ) > Ai(Kni ,...,ni,...,nj ,...,np ) . Proof. Let Ai, x and E denote, respectively, the spectral radius, the corresponding eigenvector, and the edge set of Kni ..,ni..,nj.,,np , and let Ai, A* and E* denote, respectively, the spectral radius, the adjacency matrix, and the edge set of Kni,...,ni-i,...,nj+it...np . From the Variational theorem we have A* > x A. x — ^ ^ 2xu,x uv * uv uveE* uveE uveE*\E uveE\E = Ai + 2xi(ni — 1)xi — 2xinj xj = Ai + 2xlx(A^1 — (2.2) V Ai + ni Ai + nj 1 by Eq. (2.1). Next, note that Kni,...,ni,...,nj,...,np has Kni,nj as an induced subgraph, so that, by the Interlacing theorem [2], Ai > y/npnj > nj . Therefore, ni — 1 nj (ni — nj — l)Ai — nj ^ Ai — nj ^ Ai + ni Ai + nj (Ai + ni)(Ai + nj) _ (Ai + ni)(Ai + nj) so that Ai > Ai follows from Eq. (2.2). □ Proof of Theorem 1.6 Let Kmi ,...,mp be the complete multipartite graph with the smallest spectral radius. If there are two parameters mi > m j > 2, then (mi +1) — (mj — 1) > 2 and from Lemma 2.1 Ai (Kmi ,...,mi,...,mj ,...,mp ) > Ai (Kmi !...lmi+i!...lmj — i,...,mp ) contradicting the choice of Kmi,...,m.,...,mj,...,mp . Hence, all parameters mi,..., mp are equal to one, except for one parameter equal to n—p+1, so that Kmi,...,mp = CS(n,p—1). □ Proof of Theorem 1.7 Let Km i ,...,mp be the complete multipartite graph with the largest spectral radius. It is apparent from Lemma 2.1, that |mi — mj | < 1 holds for all i = j, as otherwise, assuming mi — mj > 2, Ai (Kmi ,...,mi — i,...,mj + i,...,mp ) > Ai (Kmi ,...,mi,...,mj ,...,mp ) contradicting the choice of Kmi..,mi..,mj,...,mp . The condition |mi — mj | < 1 for i = j implies that each parameter mi is equal to either |_n/pj or [n/p], so that Kmii...,mp = T (n,p). □ Remark 2.2. Delorme [5] proved (a bit too concisely) that changing the arbitrary e parameters of a complete multipartite graph by their average value increases the spectral radius by relying on the characteristic polynomial. While this result can be substituted for Lemma 2.1 in the proofs of Theorems 1.6 and 1.7, Lemma 2.1 is an independent result based on the principal eigenvector and inspired by the rotation lemma from [1]. Remark 2.3. Delorme also asked in [5] whether the spectral radius of complete multipartite graph Knii...,np is a concave function on the (p — 1)-dimensional simplex Y : J2p=i ni = n Л (Vi G {1,... ,p})(ni > 0), i.e., whether Ai(Ki(ni,...,np) + (i —t)(mi,...,mp)) > t Ai(Kni,...,np ) + (1 — t) Ai(Kmi,..,mp ) for any two points (ni,..., np), (mi,..., mp) G Y and each t G [0,1] such that t(ni,. .., np) + (1 — t)(mi, ..., mp) G Y? Delorme proved this affirmatively for p < 3 in [5]. We tested it computationally for t = 0.5, p g {4,..., 10} and n < 33 and found no counterexamples to the above question on these simplices. References [1] F. Belardo, E. M. Li Marzi, S. K. Simić, Some results on the index of unicyclic graphs, Linear Algebra Appl. 416 (2006), 1048-1059. [2] D. CvetkoviC, M. Doob, H. Sachs, Spectra of Graphs—Theory and Application, Academic Press, New York, 1980. [3] D. Cvetkovic, P. Rowlinson, The largest eigenvalue of a graph: A survey, Lin. Multilin. Algebra 28 (1990), 3-33. [4] D. Cvetkovic, P. Rowlinson, S. Simic, An Introduction to the Theory of Graph Spectra, Cambridge Univ. Press, Cambridge, 2010. [5] C. Delorme, Eigenvalues of complete multipartite graphs, Discrete Math. 312 (2012), 25322535. [6] I. Gutman, The energy of a graph: Old and new results, in: A. Betten, A. Kohnert, R. Laue, A. Wassermann (eds.), Algebraic Combinatorics and Applications, Springer, Berlin, 2001, 196211. [7] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012. [8] M. M. Petrovic, The spectrum of infinite complete multipartite graphs, Publ. Inst. Math. (Belgrade) (N.S.) 31(45) (1982), 169-176. [9] J. H. Smith, Some properties of the spectrum of a graph, in: R. Guy, H. Hanani, N. Sauer, J. Schönheim (eds.), Combinatorial Structures and Their Applications, Gordon & Breach, New York, 1970, 403-406. [10] P. Turan, On an extremal problem in graph theory, Mat. Fiz. Lap. 48 (1941), 436-452. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 115-124 Laceable knights Michael Dupuis Univ. of St. Thomas, St. Paul, Minnesota, USA Stan Wagon Macalester College, St. Paul, Minnesota, USA Received 12 December 2012, accepted 17 August 2013, published online 26 August 2014 A bipartite graph is Hamilton-laceable if for any two vertices in different parts there is a Hamiltonian path from one to the other. Using two main ideas (an algorithm for finding Hamiltonian paths and a decomposition lemma to move from smaller cases to larger) we show that the graph of knight's moves on an m x n board is Hamilton-laceable iff m > 6, n > 6, and one of m, n is even. We show how the algorithm leads to new conjectures about Hamiltonian paths for various families, such as generalized Petersen graphs, I-graphs, and cubic symmetric graphs. Keywords: Hamilton-laceable, generalized Petersen graphs, Hamilton-connected, Hamiltonian paths, knight graph, traceable. Math. Subj. Class.: 05C45, 05C85 1 Introduction Let Nm,n be the graph of knight's moves on an mxn chessboard. Knight's tours, which are Hamiltonian cycles in these graphs, have been considered for over 1000 years and in 1991 Allen Schwenk ([13]; see also [5]) characterized the knight graphs having a Hamiltonian cycle. Assuming m < n, he proved that Nm,n is Hamiltonian iff at least one of m, n is even, m is not 1, 2, or 4, and (m, n) is not (3,4), (3, 6), or (3, 8). Two related questions arise: Which knight's graphs are (a) Hamilton-laceable (HL); (b) traceable? A bipartite graph is HL if for any two vertices in different parts there is a Hamiltonian path from one to the other. And a graph is traceable if it has at least one Hamiltonian path. An HL graph with at least one edge is necessarily Hamiltonian. Here we give a complete characterization of the HL knight's graphs: Nm n is HL iff m and n are at least 6 and E-mail addresses: mdupuis@osii.com (Michael Dupuis), wagon@macalester.edu (Stan Wagon) Abstract at least one of m, n is even. The problem has been investigated for square boards; Conrad et al. [5] showed that Nn,n is HL iff n is even and n > 6. For traceability, the problem has been solved, though unpublished. Mark Krusemeyer settled it in 1972 (see also [8]). A paper in 1978 by Cull and DeCurtins [6] settles all cases except N3 n and N4 n. The failure of the 4 x 4 case appears in [16, p. 51]. We include here a simple approach to the 3 x n and 4 x n cases. The complete characterization is that the only nontraceable, connected knight graphs are N3,5, N3,6, and N4,4. Our approach uses computation and the algorithms used have broad application to Hamiltonian properties. We discuss some conjectures inspired by connectivity and lace-ability computations on several thousand graphs. 2 Knight graph laceability Our method is similar to that of [5] in that we use computation for small cases and then settle the general case by decomposition into small cases. Using Mathematica's Find-HamiltonianCycle function for the computer search means that the amount of programming is quite small. We use HP to abbreviate Hamiltonian path. Theorem 2.1. Km,n is not HL if m < 5 or both m, n are odd. Proof. By Schwenk's Theorem we need only consider K3,n and K5,n. Those cases follow from the following lemma since the corner vertices satisfy the hypothesis. □ Lemma 2.2. If G is bipartite and Hamiltonian and has two vertices of degree 2 that have exactly one common neighbor, then G is not HL. Proof. Let u and v be the degree-2 vertices with common neighbor w. Let x be the neighbor of u that is not w and let a be a neighbor of x that is not u. Such must exist because the graph is Hamiltonian, and so has no degree-1 vertices. Then a is neither w (because graph is bipartite) nor v (w is the only common neighbor of u and v). There can be no Hamiltonian path from a to w because once a path strikes one of u, v it must go to w and therefore end without visiting both u and v. □ The main tool for the general result is a decomposition lemma, but there is a complication. Let S and F denote the start and finish points of an HP. There is one case — where S and F occupy the rightmost corners of a component board as shown in Figure 1 — where it is not clear that combining two HL boards leads to an HL board. The lemma that follows avoids this case. We say that an m x n board is HL if Km,n is HL; the width of an m x n board is n, the horizontal dimension. Lemma 2.3 (Decomposition Lemma). Suppose the m x n board G splits into two HL boards by a vertical line. Then G is HL provided • m is odd; or • m is even, n is even, and the two component boards have different widths; or • m is even, n is odd, and the two component boards have widths that differ by 2 or more. 5 F Figure 1: A difficult case on a 6 x 12 board. m F A C Л Л- Б- V У- D nk Figure 2: Building an HP from S to F. Dashed arrows stand for a longer part of a Hamil-tonian path. Proof. Note that m > 6 by Theorem 2.1, so there are at least three white and three black squares in any column. Call the two smaller boards (called halves), each with m rows, left and right. If S and F are in different halves, just take an HP in S 's half to an end point near the border, cross over to the F 's half, and finish by using an HP in that half to reach F. Suppose S and F are in the same half, say the left, and C and D are the rightmost corners of the left half. If one of C or D is neither S nor F, say C, find an HP in the left from S to F. The path must, just before reaching C, visit A or B, the two neighbors of C. Assume, by switching S and F if necessary, that it first passes through B, the rightmost of A and B. Then break at B, go into the right side to X (see Figure 2), and traverse the right side by an HP from X to Y. Finish by jumping to C and continuing to A and F by the original path. This leaves the troublesome case that S and F occupy the corners C and D; here it is not clear how to build an HP in the large board. Because S and F have opposite colors, m is even. So we can simply divide G vertically but in the other order, leaving S and F where they are. Because of the conditions on the widths of the halves, S and F no longer occupy the two rightmost corner positions of the left half, and the method of the preceding Figure 3: A resolution of the single troublesome case for K6,12. paragraph applies. □ The next result is based on an algorithm for finding HPs. Several of these cases were first done in [5], and even before that Schwenk worked out the K8,8 case by hand. Using Mathematica, the verification of all 18 cases took about 24 minutes. Theorem 2.4. Km,n is HL whenever (m, n) G {(7,8), (7,10), (9,10)} or m is 6, 8, 10, or 12 and 6 < m < n < 13. Proof. Assume (proof by computation follows) that K6,6, K6,8, and K6,ю are HL. Then their doubles (transposing as necessary) — 6 x 12, 8 x 12, 10 x 12 — can be handled by dividing the board in half vertically. Laceability follows from the proof of the decomposition lemma except in the case that S and F occupy the two right-hand corners of the left half. But those cases are easily handled computationally by a search for a single HP, using the algorithm described shortly; the case of 6 x 12 is shown in Figure 3. And having K6,6, we can resolve K 12,12. This is because the 12 x 12 board can be bisected vertically into 12 x 6s, and the decomposition proof takes care of all cases except S and F occupying the rightmost corners of the left half. But then we can divide the board horizontally into two 6 x 12s, which places S and F in different halves and allows the proof of the decomposition lemma to be applied. Also, having 6 x 13 (proof below) yields 12 x 13 by direct application of decomposition. The remaining 18 cases are 6 x 6, 6 x 7, 6 x 8, 6 x 9, 6 x 10, 6 x 11, 6 x 13, 7 x 8, 7 x 10, 8 x 8, 8 x 9, 8 x 10, 8 x 11, 8 x 13, 9 x 10,10 x 10,10 x 11, and 10 x 13. These are handled by finding a complete set of HPs. The square cases were done in [5], but we can take care of all cases with about 18 minutes of computation. Here is an algorithm based on Mathematica's fast FindHamiltonianCycle function. Hamiltonian path algorithm Input: Graph G and two vertices S, F. Output: TRUE if there is a Hamiltonian path in G from S to F ; FALSE otherwise. Step 1. Form graph H from G by adding a new vertex v and the edges v ^ S and v ^ F. Step 2. Use a Hamiltonian-cycle finder to check if such a cycle exists in H. This algorithm is quite simple and settles the 18 cases of Theorem 2.4 in 24 minutes. There is no need to apply it to all pairs (S, F) from the two parts. One can restrict S to orbit representatives under the automorphism group. Using a fast algorithm for graph isomorphism, one can add leaves to each of two vertices to determine if they are in the same orbit. When working with larger graphs having a lot of symmetry, this use of orbit representatives cuts down the computation time dramatically. □ The algorithm above was suggested by a referee. Our original algorithm was a little more complicated and can provide more speed in some cases. One can add edge S ^ F to the graph if it is not there already, and then consider, one at a time, the other edges leaving S, checking to see if there is a Hamiltonian cycle that uses that edge. By putting a time constraint on this search one avoids possible dead ends without examining all the possibilities. For many HC or HL graphs this leads to a faster algorithm; in essence, it is a way of tweaking whatever Hamiltonian cycle finder one is using. Another idea that is sometimes useful when the time limit runs out without a path being found is to permute the graph and try again. The idea of using permutations to speed up graph algorithms has been fruitful in other areas, such as the 4-coloring of planar graphs efficiently [8]. The Hamilton cycle algorithm used by Mathematica has several cases, but a main one is the Angluin-Valiant heuristic. For larger problems one can use traveling salesman problem software to search for Hamiltonian cycles (see §5). Theorem 2.5. Nm,n is Hamilton-laceable iff 6 < m, 6 < n, and one of m, n is even. Proof. The forward direction follows from Theorem 2.1. For the other direction, assume m < n. Case 1. m is odd and m < 13. In this case n is even. If n < 10, we have the conclusion by computation (Theorem 2.4), so assume n > 12. Then n splits into the even numbers 6 and n - 6, and the conclusion follows from the decomposition lemma by induction and Theorem 2.4. Case 2. m is even and m < 12. If n < 13, we have the conclusion by Theorem 2, so assume n > 14. Then n splits into 6 and n - 6; because (n - 6) - 6 > 2, the decomposition lemma and induction yield the result. Case 3. m is odd and m > 15. In this case n is even. Use induction on m; the cases 7 X n, 9 X n, 11 X n, and 13 x n follow from Case 1. For m > 15, an m x n board splits into 6 X n and (m - 6) x n boards. The first is HL by case 2 and the second is HL by the inductive hypothesis. Because (m - 6) - 6 > 2, the decomposition lemma applies. Case 4. m is even and m > 14. An m x n board splits into 6 x n and (m - 6) x n boards; because (m - 6) - 6 > 2, the result follows inductively from case 2 (for the base case) and the decomposition lemma. □ A dynamic demonstration at [15] shows all 322 paths in the case of the traditional 8 x 8 board; Figure 4 shows a typical case. As noted, this classic chessboard case was resolved in [1]; the work here settles all rectangular boards. 3 Traceable knights A graph is traceable if it has a Hamiltonian path. Cull and De Curtins [6] proved that Nm n is traceable if m > 5. Mark Krusemeyer reports having solved the complete problem in Figure 4: A snapshot (start at lower left, finish in center of second row) of a demonstration [15] that shows all 1024 knight's paths on a chessboard. Figure 5: A knight's path from (1,1) to (6, 2) on a 3 x 7 board. Similar paths exist with 7 replaced by 8 through 13, and hence for widths past 7. 1972 in an addendum to his doctoral thesis. We present here an approach to the m = 3 and m = 4 cases. For m = 3, N3,3 is disconnected, while the HP algorithm shows that N3,5 and N3 6 are not traceable and N3,n is traceable when n g {4, 7,8,9,10,11,12}. Moreover, for the last six cases, there is an HP that goes from the lower-left corner to the next-to-last rightmost point of the center row (Figure 5). Therefore it is a simple matter to chain two such paths together to get an HP in the case of n > 13. This settles the m = 3 case. The case of N4 n is harder. Using the HP algorithm with some edges deleted led to the gadget in Figure 6, which is the key to a proof that HPs exist for all N4,n Theorem 3.1 (Krusemeyer, Cull, De Curtins). The only connected knight graphs that are not traceable are N3,5, N3,6, N4,4. Proof. Cull and De Curtins [6], using methods similar to the methods of the present paper (decomposition and induction), showed that Nm,n is traceable when 5 < m < n. The case m = 3 was discussed above. A computer search shows that N4,4 is not traceable; this can also be proved by hand [16, p. 51]. Figure 6(b) shows N4,5, N4,6, and N4,7 with HPs from the lower left corner to the upper left corner. The gadget of Figure 6(a) can be used to extend any such path leftward to one with the same property, but on a board with three Figure 6: (a) The two disjoint paths in the 4 x 3 board allow one to construct Hamiltonian paths in any 4 x n board with n > 5. (b) Corner-to-corner paths for 4 x 5, 4 x 6, 4 x 7. more columns. This gives corner-to-corner HPs for all (4, n) with n > 5. The gadget was found by using the HP algorithm on a larger graph with various edges deleted so to force the condition of one edge out and one edge in. □ 4 Further applications of the algorithm The HP algorithm can be used on many families of bipartite graphs and works as well to study whether a nonbipartite graph is Hamilton-connected (HC), which means that there is a Hamiltonian path between any two vertices. Mathematical GraphData database has 572 graphs that are bipartite, Hamiltonian, vertex-transitive, and not a cycle; the largest has 2048 vertices. Using the HP algorithm we have shown that all 572 are HL. In particular, the HP algorithm shows that all the Foster graphs (connected, cubic, edge-transitive, and vertex-transitive) with 768 or fewer vertices (as well as the known Foster graphs up to 1000 vertices) are HL when bipartite and HC otherwise. Recall the folklore conjecture (based on a question by Lovasz about traceabiliy of vertex-transitive graphs; see [10]) that connected vertex-transitive graphs are Hamiltonian except for five small examples. Computations show that one almost always gets stronger Hamiltonian properties than just the existence of a Hamiltonian cycle. Question 4.1. Are even cycles the only bipartite, Hamiltonian, vertex-transitive graphs that are not Hamilton-laceable? And the same behavior has been observed for edge-transitive graphs. Question 4.2. Are even cycles the only bipartite, Hamiltonian, edge-transitive graphs that are not Hamilton-laceable? One can ask similar question for Hamilton-connected graphs. Of the 1557 nonbipartite, vertex-transitive, Hamiltonian graphs in the database (cycles excluded), the HP algorithm resolved the Hamilton-connected status of all of them. Only one — the dodecahedral graph — failed to be Hamilton-connected. Several of these graphs had over 1000 vertices; the largest was Kneser14,6 with 3003 vertices. Question 4.3. Are the odd cycles and the dodecahedral graph the only nonbipartite, Hamil-tonian, vertex-transitive graphs that are not Hamilton-connected? And we have the same question with edge-transitivity replacing vertex-transitivity. Here are more detailed summaries for some interesting families: Generalized Petersen graphs GP(n, k). These cubic graphs, generalizations of the Petersen graph, are defined when n > 3 and k < n/2. They are all Hamiltonian except GP(6q + 5,2) and isomorphs (Alspach [1]). These are bipartite when n is even and k is odd. It is easy to see that GP(n, 1) is HL when bipartite, HC otherwise. Alspach and Liu [2] have shown that GP(2m, 3) is HL; some authors include cases such as GP(5, 3), which is isomorphic to the non-Hamiltonian Petersen graph GP(5, 2), but here we restrict to k < n/2. A recent paper [12] by Richter contains further results on Hamiltonian paths in this family. The HP algorithm shows that GP(n, k) is HL when n < 100 and so it is reasonable to conjecture that all bipartite graphs in the family are HL. Alspach and Qin [3] proved that GP(4m, 2m - 1) is HL. The situation regarding HC and HL for k < 3 is: GP(n, 1) is HL when n is even, HC otherwise (Alspach and Liu [2]). GP(n, 2) is HC iff n = 1 or 3(mod 6) [2, 12] GP(n, 3) is HL when n is even, HC otherwise [2]. Pensaert [11] found that GP(6,2) and GP(12,4) are not HC and conjectured that GP(n, k) is HC or HL whenever n > 3k +1. The case GP(3k + 1, k) is isomorphic to GP(3k + 1,3), a settled case. The HP algorithm extends the Pensaert examples by showing that GP(6q, 2q) is not HC when q < 12. The algorithm also shows that, with the exceptions mentioned for bipartite, non-Hamiltonian (GP(6q + 5,2)), and GP(6q, 2q), the graphs GP(n, k) are HC for k < 16 and 2k +1 < n < 100. So we have the conjectures that these last two results, positive and negative, hold in all cases. The I-graph computations discussed next show that GP(n, k) is HL in all the bipartite cases with n < 100. I-graphs I(n,j,k). These cubic graphs, where 1 < j, k < n and j, k = n/2, are generalizations of the GP family; GP(n, k) = I(n, 1, k) when k < n/2. An I-graph is bipartite and connected iff n is even, j and k are odd, and gcd(n, j, k) = 1. Horvat et al [7] give a simple condition for two I-graphs to be isomorphic. Computation suggests that all connected I-graphs, except the generalized Petersen exceptions, are Hamiltonian; that has been recently proved by Bonvicini and Pisanski [4]. Even stronger Hamiltonian properties appear to hold: the HP algorithm can be used to check for laceability: all bipartite, connected I(n, j, k) for n < 100 are HL. Bipartite Kneser graphs Hn,k. It is a well-known conjecture that the connected ones (k = n/2) are Hamiltonian; this has been proved for n < 27 [14]. These graphs are vertex- and edge-transitive, so it suffices to restrict the start point for a Hamiltonian path to a single vertex, v. Further, using permutations on the underlying n-set shows that one need only consider k + 1 types of pairs (v, F), the type being the size of the intersection of the set represented by v with that represented by F. These considerations speed up the HP algorithm, and the results are that Hn,k is HL for all 4 < n < 14 and k < n/2, and also for (n, 2) with n < 43 and (n, 3) with n < 25. Note that H3,1 is a 6-cycle and so is not HL. 5 Conclusion The work presented here shows how one can call on fast algorithms even for NP-complete problems as a way of learning about families of graphs, so long as the graphs are not terribly large. There are 853 bipartite Hamiltonian graphs in Mathematica's database and the algorithms presented here have resolved the laceability of all of them. Since finding Hamiltonian cycles or paths is NP-complete, it is not surprising that our methods fail for some large graphs. An idea that has proved useful on the larger examples (such as Kneseri3,6, Kneser14,5, Kneser14,6, and an alternating group Cayley graph with 2520 vertices) is to use the cutting-edge technology of traveling salesman problem software, such as Concorde or LKH. One can turn a graph into a weighted complete graph with weights 0 on the edges and 1 on the nonedges; then if a shortest traveling salesman tour has weight 0, one has a Hamltonian cycle; if not, then no such cycle exists. This idea was used to determine Hamilton-connectivity in the large examples. Acknowledgments We are grateful to Brian Alspach and Bruce Richter for valuable comments, to Rasmus Kamper for helping with the setup of Concorde and LKH, and to referees for several helpful comments. References [1] B. Alspach, The classification of Hamiltonian generalized Petersen graphs, Journal of Combinatorial Theory Series B 34 (1983), 293-312. [2] B. Alspach and J. Liu, On the Hamilton connectivity of generalized Petersen graphs, Discrete Mathematics 309 (2009), 5461-5473. [3] B. Alspach and Y. Qin, Hamilton-connected Cayley graphs on Hamiltonian groups, Europ. J. Combinatorics 22 (2001), 777-787. [4] S. Bonvicini and T. Pisanski, Classification of Hamiltonian I-graphs and good walks in certain graph bundles, preprint 2012. [5] A. Conrad, T. Hindrichs, H. Morsy, and I. Wegener, Solution of the knight's Hamiltonian path problem on chessboards, Discrete Applied Mathematics 50 (1994), 125-134. [6] P. Cull and J. De Curtins, Knight's tour revisited, Fibonacci Quarterly 16 (1978), 276-285. [7] J. P. Hutchinson and S. Wagon, Kempe revisited, American Mathematical Monthly 105 (1998), 170-174. [8] R. Gaebler and T.-W. Yang, Knight's tours, unpublished notes 1999, http://gaebler. us/share/Knight_tour.html. [9] B. Horvat, T. Pisanski, and A. Zitnik, Isomorphism checking of /-graphs, Graphs and Combinatorics 28 (2012), 823-830. [10] K. Klavdija and D. Marušic, Hamilton cycles and paths in vertex-transitive graphs — current directions, Discrete Mathematics 309 (2009), 5491-5500. [11] W. P.J. Pensaert, Hamilton paths in generalized Petersen graphs, Masters Thesis, Univ. of Waterloo, (2002), http://etd.uwaterloo.ca/etd/wpjpensaert2 0 02.pdf. [12] B. Richter, Hamilton paths in generalized Petersen graphs, preprint, June 2012. [13] A. Schwenk, Which rectangular chessboards have a knight's tour?, Mathematics Magazine 64 (1991), 325-332. [14] I. Shields and C. D. Savage, A note on Hamilton cycles in Kneser graphs, Bulletin of the Institute Combinatorics and its Applications 40 (2004), 13-22. [15] S. Wagon, Laceable knight graphs, Wolfram Demonstrations Project (2012) http://demonstrations.wolfram.com/LaceableKnightGraphs [16] J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton Univ. Press, Princeton, New Jersey, 2004. ars mathematica contemporanea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 9 (2015) 125-144 Matrices and their Kirchhoff graphs Joseph D. Fehribach * Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609-2247, USA Received 11 October 2012, accepted 21 July 2014, published online 28 September 2014 The fundamental relationship between matrices over the rational numbers and a newly defined type of graph, a Kirchhoff graph, is established. For a given matrix, a Kirchhoff graph represents the orthogonal complementarity of the null and row spaces of that matrix. A number of basic results are proven, and then a relatively complicated Kirchhoff graph is constructed for a matrix that is the transpose of the stoichiometric matrix for a reaction network for the production of sodium hydroxide from salt. A Kirchhoff graph for a reaction network is a circuit diagram for that reaction network. Finally it is conjectured that there is at least one Kirchhoff graph for any matrix with rational elements, and a process for constructing an incidence matrix for a Kirchhoff graph from a given matrix is discussed. Keywords: Kirchhoff graphs, fundamental theorem of linear algebra, reaction networks. Math. Subj. Class.: 05C20, 05C50, 05C90 1 Introduction To understand the relationship between matrices and a newly-defined type of graph, a Kirchhoff or fundamental graph, consider the simple directed graph in Figure 1 and the incidence matrix for this digraph: In this matrix element ajj = 1 if the j-th edge s j exits the i-th vertex v j, ajj = — 1 if s j enters v j, and a j =0 if s j is not incident to v j. This use of 1 and —1 is the opposite of what many authors use in graph theory, but it matches the forward direction for steps *The support of the National Science Foundation under grant DMS-0707692 is gratefully acknowledged. Website: http://www.wpi.edu/\textasciitildebach E-mail address: bach@math.wpi.edu (Joseph D. Fehribach) Abstract 101 1 1 0 011 (1.1) in chemical reaction networks and hence is preferred here. The edges of the graph in Figure 1 and indeed of all the graphs considered here are vectors (have specified length and direction). Notice that the rows of this incidence matrix are linearly dependent, and that the vector [1,1, -1]T is in its null space. But this null-space vector also represents the cycle1 in this digraph since (1)si + (1)S2 + (-1)s3 = 0 is the cycle. Also the cuts for each vertex2 of this graph are (as always) represented by the rows of the incidence matrix. So the graph in Figure 1 is a representation of the orthogonality of the null and row spaces of this incidence matrix, and this orthogonality is essentially v 2 v v s 3 3 Figure 1: A simple example to illustrate the concept of Kirchhoff graph. the classic result that the cycle space and the cut space of a (standard) graph are orthogonal complements (cf. e.g. Diestel(1997) [5, p. 22] or Bollobas(1998) [1, p. 53]). The relationship between the graph in Figure 1 and its incidence matrix easily extends to multi-digraphs where the edges are again vectors. As a simple example of this extension, consider the multi-digraph in Figure 2 and its incidence matrix3: to 0 1 2 0 1 to 1 0 -2 1 0 0 -1 -1 = 0 -1 -1 0 1-1 0 0 0 0 0 0 1-1 0 0 0 Here "1 - 1" means that the same vector enters and exits a given vertex. Again the graph in Figure 2 represents the orthogonality of the row and null spaces of this incidence matrix since the cycle is (1)si + (2)S2 + (-2)s3 = 0 and the vertex cuts "lie" in this row space. What is really interesting is that the relationship can be extended even further: For any matrix with rational elements, it would seem possible to construct a multi-digraph whose 1For the current discussion, a cycle in a graph is a closed walk, i.e. an alternating sequence of vertices and edges incident to adjacent vertices {vi, si, v2, ...vn, sn, v1} beginning and ending with the same vertex, with no assumption that any vertices or edges are distinct. 2A vertex cut in a graph is the set of edges incident to the vertex (the minimum set of edges that would need to be removed to isolate the vertex from the rest of the graph). 3The term incidence matrix as used here is a generalization of the standard definition. Here each column corresponds to an edge vector, even though that vector may appear multiple times in the graph Figure 2: A multi-digraph that is a somewhat more complicated Kirchhoff graph. The double hash marks crossing s1 indicate that two copies of this edge vector connect the first two vertices. cycle space corresponds to the null space of the matrix and whose cut space lies in the row space of the matrix. Indeed the concept of a Kirchhoff graph is a sort of inverse of the relationship between the cycle space and the cut space for the standard graph in that one starts with an arbitrary matrix and then attempts to construct the graph. Thus each Kirchhoff graph for a matrix is a graph-theoretic representation of the fundamental theorem of linear algebra which states that the null space and the row space of any matrix are orthogonal complements.4 Because of the relationship between Kirchhoff graphs for a given matrix and the fundamental spaces for that matrix, a Kirchhoff graph can also be termed a fundamental graph. Also from the simple examples above, one sees that the exact length and orientation of the edge vectors are not important. The key issue, rather, is the multiplicities of any given vector in a cycle and the multiplicities of that vector between given vertices. The graph in Figure 1 is a Kirchhoff graph for any matrix with the same row space and null space as the matrix in (1.1), and the graph in Figure 2 is a Kirchhoff graph for any matrix with the same row space and null space as the matrix in (1.2). The concept of a Kirchhoff graph comes out of chemical reaction network theory. As the name implies, when based on a reaction network, a Kirchhoff graph satisfies both the Kirchhoff current law and the Kirchhoff potential law, and is therefore a circuit diagram for that reaction network. Their role in this context was discussed by Fehribach(2009) [9], and also by Fishtik, Datta et al. [11, 12, 13, 10, 14, 25] and in some of their references. In the latter works, Kirchhoff graphs are referred to as reaction route graphs. The concept of a Kirchhoff graph thus connects the fundamental theorem of linear algebra with the fundamental conservation principle in the Kirchhoff laws. There is also the important and distinct concept of a Kirchhoff or Laplacian matrix and the well-known Kirchhoff theorem which relates the eigenvalues of the Kirchhoff matrix for a graph to the number of spanning trees of that graph. A variety of other graphs have been used to discuss reaction networks. For a general review, particularly of early uses, see Fehribach(2007) [8]. An important recent sequence of work on graphs and reaction networks began with the work of Horn(1973) [15, 16], followed by that of Perelson & Oster(1974) [22], Clarke(1980) [3], Schlosser & Fein-berg(1994) [23], and the work of Craciun & Feinberg(2006) [4]. The graphs studied in 4Not all authors/texts use this terminology, but it has become more widely used in recent years; cf. Strang(2003) [24]. these articles are termed species-complex-linkage (SCL) graphs and species-complex (SC) graphs, and are useful in studying reaction kinetics and the stability of equilibria, but they are not connected to the fundamental spaces of any matrix, nor do they give a representation of the fundamental theorem of linear algebra. Also in general the above articles specifically discuss graphs; see their references for other related articles that discuss the reaction networks themselves. Throughout this work, we consider matrices with rational elements, but then consider basis vectors for the null and row spaces with integer elements. This is possible since for any matrix over the rationals, one can multiply each element by the least common multiple of all the denominators to obtain a matrix with integral elements, but the same null and row space as the original rational matrix. Also it is important to realize that the correspondences between the null and row spaces on the one hand and the cycle and cut spaces on the other are not equivalences. The null and row spaces are vector spaces over the rationals, while the cycle and cut spaces are modules over the integers (it is easy to interpret integral multiples of a cycle, but not fractional multiples). For a Kirchhoff graph G of a matrix A, there are natural embeddings of the cycle space of G into the null space of A, and of the cut space of G into the row space of A. The next section contains the formal definition of a Kirchhoff graph and several basic results; these make rigorous the ideas introduced in the examples above. Since the impetus for the concept of Kirchhoff graph comes from the theory of chemical reaction networks, Section 3 presents the development of a Kirchhoff graph for such a network. The final two sections discuss the construction of Kirchhoff graphs, as well as open problems and future work. 2 Kirchhoff Graphs—Definition and Basic Results Consider an m x n matrix over the rationals: A G Mm,n(Q). The main question to be considered here is "Which graph or graphs best represent this matrix in terms of its fundamental spaces and the fundamental theorem of linear algebra?" That is, which graph(s) best reflect the duality that Row(A) and Null(A) are orthogonal complements: Row(A) ± Null(A), RowT(A) e Null(A) = Qn where the superscript T indicates the transpose of all vectors in the space. The proposed answer to this question is the Kirchhoff graph: Definition 2.1. For a given matrix A G Mm,n(Q), a geometric cyclic multi-digraph G is a Kirchhoff graph for A if and only if the following two conditions are satisfied: 1. For u j G Z, u = [ui,M2, ...,un]T G Null(A) if and only if there is a cycle in G where, for each j, 1 < j < n, the jth directed edge appears with multiplicity |uj |. The sign of u j gives the relative direction that the jth edge is crossed. 2. For a given vertex of G, if the jth edge exits with multiplicity v j G Z, then v = [v1, v2,..., vn] G Row(A). When v j is negative, the edge enters the vertex (exits in the negative sense). Remark 2.2. 1. While in general A e Mm,n(Q), there is no loss of generality in assuming that A e Mm,n(Z) since one can multiply A by the least common multiple of all the denominators of the elements of A without affecting the null and row spaces. Similarly one can take the elements of the basis vectors for the null space or row space to be integers. 2. According to this definition, two matrices with the same null and row spaces have the same Kirchhoff graph(s). 3. For the present discussion, a digraph G is cyclic if and only if Sj is an edge vector of G implies Sj is an edge vector of some cycle C с G. The trivial graph with one vertex and no edges is then vacuously cyclic. 4. These graphs are "geometric" in that edges are vectors. Two edges with the same length and direction are the same vector and are therefore identified. For discussions of the more-general topic of geometric graphs, see Pach, et al. [19, 20, 21]. 5. The second condition in Definition 2.1 assures that all of the vertex cuts lie in the row space of the matrix, but not all dimensions of the row space necessarily need to be represented explicitly as vertex cuts. As is discussed below, if A is nonsingular, the simplest Kirchhoff graph is a single vertex with no edges. 6. Making the outward direction positive matches the tradition in analysis and is what is needed for discussing reaction networks; unfortunately it is the opposite what is generally used in graph theory. 7. In the extreme cases where either Null(A) = {0} or Row(A) = {0}, the Kirchhoff graph can be defined as a single vertex with no edges. When Null(A) = {0}, this graph results from the only allowed cycle being a null cycle where all edge vectors appear an even number of times and cancel as one moves around the cycle; the simplest null cycle is a single vertex. When Row(A) = {0}, then the second condition in the definition requires that all edges vectors begin and end at the same vertex and therefore have length zero. 8. As will be discussed in Section 3 below, if the matrix A is the transpose of the stoichiometric matrix for a reaction network, the first condition in Definition 2.1 is a form of the Kirchhoff potential (or voltage) law, while the second condition is a form of the Kirchhoff current law. The rest of this section is devoted to some basic properties and results associated with Kirchhoff graphs. 2.1 Some Basic Results Proposition 2.3. Suppose that A e Mm,n(Q) and Null(A) = Span{[ai, a2,..., an]T}, aj e Z, with at least one aj = 0. Then dim(Null(A)) = 1 and dim(Row(A)) = n — 1, and a Kirchhoff graph for A can be given as a single cycle with |a11 + |a2| + ... + |an| vertices. Proof. Suppose first that Null(A) = Span{[a1, a2,..., an]T} with aj = 0 Vj. Then Row(A) = Span{[—a2, a1,0,..., 0], [0, — a3,a2,..., 0],..., [0,..., —an,an-1]}, and these n — 1 vectors give n — 1 cut (vertex balance) conditions for vertices where differing edge vectors meet. The final condition is a linear combination these row-space basis vectors: [-an, 0,0,..., ai]. Unless |aj | = 1 Vj, there are also null vertices where the same edges enter and exit. If aj = 0 for some j, then the unit vector with 1 as the j-th element and all other elements being 0 lies in Row(A). The edge Sj does not appear in the graph, and the vertex balance conditions are formed as before leaving out this unit vector. (Recall that the definition of a Kirchhoff graph does not require that all the basis vectors for the row space be represented in a graph.) □ Example 2.4. A simple example is helpful in understanding the above proposition: Suppose that Null(A) = Span{[0,1, -3, 2]T}. Then Row(A) = Span{[1,0,0,0], [0, 3,1,0], [0,0, 2, 3]} and [0, -2,0,1] is the additional vector from Row(A) needed for the vertex where the s4 edges join s2. Figure 3 shows a one-cycle Kirchhoff graph for this example. Figure 3: A simple cycle to illustrate Proposition 2.3. The hash marks give the multiplicity of each edge indicating how the cut space of the graph corresponds to Row(A). Remark 2.5. If fewer than three of the elements aj are nonzero, then the Kirchhoff graph is degenerate, as when the null space or row space is trivial. When only one aj is nonzero, there is one vertex and an edge of length zero (the edge must begin and end at the same vertex); when two aj are nonzero, there are two vertices and two overlapping edges, as in the degenerate case shown in Figure 5 below. The previous proposition suggests that matrices can have multiple Kirchhoff graphs since the edges can appear in any order in the cycle. For example, one could split the two s4 vectors so that the order moving around the cycle is s2, s4, -3s3, and finally the second s4. In fact, a given matrix may have two or more Kirchhoff graphs which differ even more than just the order that edges appear in a cycle. Proposition 2.6. A given matrix A e Mm,n(Q) may have multiple, distinct Kirchhof graphs, i.e., a matrix does not necessarily have a unique Kirchhoff graph. Proof. Consider the matrix " 10 11 A 1101 (2.1) Here Row(A) is just the span of the two rows of A, while Null(A) = Span • 1 0 1 1 —1 1 0 —1 (2.2) At least5 two Kirchhoff graphs exist for this matrix (and any other matrix with the same row and null spaces), as seen in Figure 4. □ Figure 4: Two Kirchhoff graphs for A in (2.1). Again the hash marks in Graph 2 indicate that two copies of that edge are needed in that position. Now suppose that dim(Row(A)) = 1 and thus dim(Null(A)) = n — 1. This is again a degenerate case, and a Kirchhoff graph in this case again depends on what one is willing to accept as a cycle. For our purposes here, we again allow for a degenerate cycle where the edge vectors double back on themselves. Proposition 2.7. Suppose that A e Mm,n(Q) and that dim(Row(A)) = 1. Then dim(Null(A)) = n — 1, and a Kirchhoff graph for A can be given as a degenerate one-dimensional set of cycles whose edge vectors lie on top of each other. Proof. Suppose first that Row(A) = Span{[ab a2,..., an]} with a,j = 0. Then Null(A) = Span{[—a2, a1,0,..., 0]T, [0, —a3, a2,..., 0]T,..., [0, 0,..., — an, an-1]T}, and these n — 1 vectors represent n — 1 degenerate cycles. Each of the vertices in the middle of the cycle is a null vertex—a vertex where exactly the same edges enter and exit. On the other hand, the edges entering/leaving the two end vertices satisfies the row space condition, i.e., at each end |aj | copies of s j enter or exit and the sign of a j determines which end the edge s j enters and which end it exits. If aj = 0, then Sj = 0 since this is the only edge vector that ends where it begins. □ Example 2.8. Reversing the situation in the previous example, suppose that Row(A) = Span{[0,1, —3,2]}. Then Null (A) = Span{[1,0,0,0]T, [0, 3,1,0]T, [0,0,2, 3]T}. The first vector in the span ([1,0,0,0]T) implies that s1 = 0. Figure 5 shows a degenerate Kirchhoff graph for this example. 5Clearly the union of any two Kirchhoff graphs is also a Kirchhoff graph, but the author believes that these are the only two minimal or prime Kirchhoff graphs for this matrix. S4 •—//-» //->' //- S3 -VA--VA-• s 2 v v LR Figure 5: A degenerate Kirchhoff graph to illustrate Proposition 2.7. The three sets of vectors must be overlaid to form the Kirchhoff graph so that all edge vector sets begin or end at the two end vertices vL and v R. The middle vertices are null vertices where copies of an edge both begin and end. Another basic case that should be considered is the one mentioned in the Introduction: Proposition 2.9. Suppose that A e Mm,n (Q) is row equivalent to the incidence matrix for a standard cyclic digraph (having at most one edge vector between any two vertices, and no edge vector appearing multiple times in the digraph). Then this digraph is a Kirchhof graph for A. Remark 2.10. A matrix is the incidence matrix for a standard cyclic digraph if and only if (1) all elements are 0 or ±1, (2) each column has exactly one 1 and one -1, (3) no two columns have their nonzero elements in the same rows, and (4) each row has at least two nonzero elements. Proof. In this case the standard result that the cycle space and the cut space of a graph are orthogonal complements yields the desired result, since Null( A) and Row( A) are preserved under row operations (cf. e.g. Diestel [5, p. 22] or Bollobas [1, p. 53]). □ One might think that every geometric cyclic multi-digraph is the Kirchhoff graph for some matrix; this is not the case, as the following counterexample shows. Consider the Figure 6: Three similar geometric cyclic multi-digraphs: the leftmost is not the Kirchhoff graph of any matrix; the center and rightmost are Kirchhoff graphs. Note that the Kirchhoff graph in the center is not minimal; it is simply the union of two triangular Kirchhoff graphs, each triangular graph having one copy of each edge vector. leftmost geometric cyclic multi-digraph in Figure 6. If this graph is a Kirchhoff graph for some matrix A, then [1,1, — 1]T must be in Null(A) and [1,1,1] must be in Row(A) which of course is impossible since these vectors are not orthogonal. So the cycle space and the cut space for this graph are not orthogonal. There are two possible Kirchhoff graphs similar to this non-Kirchhoff graph, and they are shown in the center and on the right in Figure 6. Finally the following proposition is an immediate consequence of the definition of Kirchhoff graph: Proposition 2.11. If G is a Kirchhoff graph for a matrix A, and if B is the same as A except that columns ji and j2 are interchanged, then a Kirchhoff graph for B is obtained by interchanging the labeling of edge vectors Sj1 and Sj2 in G. 3 Kirchhoff Graphs and Reaction Networks Let us now consider connections between Kirchhoff graphs and chemical reaction networks. As was mentioned in the Introduction, the study of such reaction networks led to the definition of a Kirchhoff graph. The reaction network discussed below describes the production of sodium hydroxide from a brine (salt) solution through electrolysis. There are a number of processes to accomplish this production (Castner-Kellner, diaphragm, membrane), and a number of variations of these processes [2, 17]. The steps presented here do not all necessarily occur in all processes, but considering all the steps together allows one to compare various processes. The reaction steps for the network to be studied here are as follows: si 52 53 54 55 56 57 58 NaCl 2Cl- 2Na+ + 2e- + 2H2O H+ + Cl-Na+ + OH- 2H+ + 2e-H2O 2H2O + 2e- Na+ + Cl-Cl2 + 2e-2NaOH + H2 HCl NaOH H2 H+ + OH- H2 + 2OH- (3.1) For this network, the charged species are viewed as intermediate, while the uncharged species are viewed as terminal. The net concentrations of the intermediate species are constant as the reaction steps proceed, while the terminal species are being produced or consumed by the reaction network. Based on these steps (3.1) and this definition of intermediate and terminal species, the achievable overall reactions for this reaction network are determined using basic linear algebra [9]. In this case the two achievable overall reactions are bi : 2NaCl + 2H2O ^ Cl2 + H2 + 2NaOH b2 : NaCl + H2O ^ HCl + NaOH ( ) Combining these two overall reactions (3.2) with the reaction steps (3.1) above, one obtains the entire reaction network. Now consider the stoichiometric matrix for this network: 0 1 0 1 0 -1 0 0 0 0 0 2 -2 0 0 0 0 0 1 0 0 0 -2 0 0 -2 0 0 -2 0 1 0 2 0 -1 0 0 -1 0 0 0 0 1 0 AT = 0 0 -1 -1 0 0 0 0 0 0 1 -2 0 0 0 -2 0 0 0 1 0 0 0 0 1 0 1 0 -1 0 0 0 0 -2 0 2 0 0 0 -2 0 1 0 0 0 0 0 0 0 -2 -2 1 1 0 2 0 0 0 0 0 -1 -1 0 0 1 1 (3.3) In a stoichiometric matrix, the entries are the stoichiometric coefficients for each chemical equation with positive entries for products and negative entries for reactants (species that are consumed in a reaction step). The rows of AT correspond to the eight steps plus the two overall reactions; the columns correspond to the eleven chemical species, namely, in order, e-, Cl-, OH-, Na+, H+, NaCl, H2O, Ch, H2, HCl and NaOH. To construct a Kirchhoff graph for this reaction network, we must compute both Null(A) and Row(A) where A is the transpose of the stoichiometric matrix in (3.3); one finds that Null(A) = Spanju 1, u2, us, u4} where u 1 := 0 0 2 1 0 0 1 0 -1 0 1 0 0 0 0 1 2 0 0 1 0 , u2 := 1 , us := 0 , u4 := 0 0 2 0 1 1 -1 0 0 0 0 -1 0 0 0 0 -1 (3.4) and that Row(A) = Spanjv1, v2, vs, v4, v5, v6} where v1 : = [ 1 0 -2 0 -1 000 0 0 v2 : = [ 0 0 0 0 1 0 -1 -2 0 0 vs = [ 0 0 0 1 0 2 -1 0 0 0 v4 : = [ 1 0 0 0 0 000 2 1 v5 = [ 0 1 0 0 0 000 1 0 ve = [ 0 0 0 1 0 000 0 1 (3.5) Null space vectors u3 and u4 give the combinations of reaction steps that yield the two overall reactions; vectors u1 and u2 give two linearly independent null cycles among these steps—combinations of steps that exactly cancel. The six vectors in (3.5) give six linearly independent reaction-rate balance conditions which guarantee the rates for all ten steps in the reaction network balance so that all of the species concentrations change only through the overall reactions (see the specific example for rNaci below). By inspection, the columns and rows in (3.4) and (3.5) form a basis for Null(A) and Row(A), respectively. Figure 7: Kirchhoff graph for NaCl-NaOH network. Hash marks again indicate the multiplicity of edge vectors between vertices. One Kirchhoff graph for this network is shown in Figure 7. This Kirchhoff graph is in fact the two dimensional projection of vectors which actually lie in an eleven dimensional phase space for the eleven chemical species in this reaction network. All of the nodes of Figure 7 lie in Row(A), while the cycles of Figure 7 correspond to vectors that span Null(A); these are of course the two defining properties that make the graph in Figure 7 a Kirchhoff graph for this reaction network, (3.1) and (3.2). Indeed one can confirm that the graph in Figure 7 is Kirchhoff by observing that the cut or incidence vectors of edges for each vertex are linear combinations of the vectors in (3.5), while the cycles in this graph are linear combinations of the vectors in (3.4). To see how the Kirchhoff graph in Figure 7 satisfies the Kirchhoff laws and is thus a circuit diagram for the NaCl-NaOH reaction network, the concepts of electrochemical potential and component potential must be introduced. An electrochemical potential mX can be defined for each species X in a reaction network, and as potentials, they can be combined linearly. The linear combinations of electrochemical potentials that are based on the stoichiometry of each reaction step Sj and each overall reaction bk are the component potentials for the reaction network. Up to an arbitrary reference potential, these component potentials are the potentials for the vertices of the Kirchhoff graph. So for example, if the potential of vertex v2 in Figure 7 is set to a reference potential (pv = ), then the electrochemical potential for the vertex v3 at the opposite end of s2 is given by the stoichiometry of the second reaction step: Mv = Mref + 2MC1- — MCl2 — 2Me- The signs in the linear combination are determined by the direction of the reaction step. The difference in potential between two vertices connected by a reaction step vector is the affinity of that reaction step and is a property of that reaction step. Similarly a component potential can be determined for each vertex, and because of the stoichiometry, the net change in potential around any cycle in this Kirchhoff graph must be zero. This is of course the Kirchhoff potential law. For a more thorough discussion of electrochemical potentials and component potentials in the context of reaction networks, cf. e.g. Newman(2004) [18] andFehribach [7, 6]. To see that the Kirchhoff graph also satisfies the Kirchhoff current law, consider the reaction steps that are incident on vertex v i. Because the total amounts of each species must be conserved when the overall reactions are taken into account, the reaction rates for each reaction step (rSj and ru ) are not independent, but rather must satisfy a set of rate-balance conditions: the rate of production/consumption of species X must be zero (rX = 0). So for example, the balance for vi represents the rate (or current) balance for the consumption of NaCl. Since NaCl occurs in reactions si, òi and b2, the stoichiometry again yields the needed rate balance condition: rNaCl = rSi + 2rbi + rb2 = 0 This is the Kirchhoff current law at vertex vi and corresponds precisely to v4 in (3.5). Similar rate balance conditions for other species or linear combinations of species hold for each vertex in the Kirchhoff graph. Thus both Kirchhoff laws are satisfied by this Kirchhoff graph, and the result is a fundamental connection between the fundamental theorem of linear algebra for the transpose of the stoichiometric matrix for a reaction network and the conservation principles of the Kirchhoff laws. Since the Kirchhoff laws are satisfied, a chemical engineer can use a Kirchhoff graph as a circuit diagram for a chemical reaction network in the same way an electrical engineer can use a traditional circuit diagram for an electrical network. 4 Existence and Construction of Kirchhoff Graphs As the NaCl-NaOH network makes clear, it is relatively easy to verify that a given geometric cyclic multi-digraph is a Kirchhoff graph for a given reaction network or a given matrix. One needs only to check that the two conditions in the definition for a Kirchhoff graph are satisfied. The two real issues are those of existence and construction of Kirchhoff graph(s) for a given reaction network or matrix. The question of existence is open, but the author offers the following conjecture: Conjecture 4.1. Every matrix over Q has a Kirchhoff graph. Proving this conjecture seems to be a difficult and interesting issue. While no proof is given here, a process for constructing Kirchhoff graphs offers one possible approach for a constructive proof—showing that the process always converges to a Kirchhoff graph would establish existence. The following two subsections demonstrate this process for two of the matrices above; the first is relatively simple, the second, somewhat more complicated. The final subsection below gives a general summary of the process. 4.1 Construction of First Kirchhoff Graph in Figure 4 One method to systematically construct a Kirchhoff graph for A in (2.1) is to "weave" the bases vectors of the null space through the row space to form the incidence matrix for a graph. To begin, consider two rows of A and the first basis vector for Null(A) in (2.2). Since the first elements in these rows are 1 and -1, these rows can correspond to the first two vertices in the graph with edge s1 connecting them. Now since the second element of the first row is a zero, one should consider the negative of the second row as a new third row of the developing incidence matrix; this new row corresponds to a new third vertex, with s2 now connecting the second and third vertices. But there would seem to be a problem in that the zero in the third position of the third row does not seem to allow an edge to connect the first and third vertices or to complete the first cycle. This problem can be overcome, however, by adding the negative of the first row to the original choice for the third row; this allows the 1 in the third position of the first row and the -1 in the third position of the new third row to correspond to -s3, completing the cycle corresponding to the first basis vector for Null(A). The partial three-vertex incidence matrix constructed so far is Here bold face numbers correspond to the vertices of this first cycle; the non-bold entries in the fourth column correspond to "loose ends" that must be "tied up" as the incidence matrix is developed. Only when all of the loose ends are dealt with in some way can one hope to have the incidence matrix for a complete Kirchhoff graph. Now to tie up at least some of the loose ends and incorporate the next basis vector from Null(A), additional rows from Row(A) corresponding to new vertices must be appended to the developing incidence matrix. Since this next null space basis vector corresponds to the cycle s2 + s3 - s4, it would seem to make sense to start with the s2 edge represented in the second column of the partial incidence matrix (4.1). Moving across this edge in the forward direction, one needs a forward copy of s3; this can be created by adding the negative of the third row in (4.1) to that third row (thus the vertex corresponding to the new third row becomes a null vertex), then appending as a fourth row the negative of the first row. The resulting (still) partial incidence matrix is now The cycle corresponding to the second basis vector from Null(A) is now shown in bold in (4.2), while the entries corresponding to the remaining two edges of the first basis vector are now in italics, and the remaining loose ends are in standard typeface. The partial incidence matrix in (4.2) still has three loose ends and thus is still incomplete. But these loose ends themselves now lie in Row(A) and thus can be tied up by appending one more row which is the negative of the current second row. To make the final incidence matrix symmetric, this final new row should be the fourth, and the previous 1 0 1 1 11 0 1 0 -1 -1 -2 (4.1) 1 0 1 1 1 1 0 1 0 1-11 - 1 0 1 0 1 1 (4.2) fourth row should become the fifth. The now complete incidence matrix is 1 0 1 1 1 1 0 1 0 1-1 1-1 0 1 -1 0 -1 1 0 1 1 Again the typeface indicates which column-element pairs represent vertices which are connected by an edge. All of the vertices satisfy the second Kirchhoff graph condition because all of the rows of (4.3) are in Row(A), and it is easy to check that all of the cycles represented in (4.3) are in Null( A). Keeping in mind that an edge vector that appears multiple times in the graph must always have a fixed length and direction, one can use (4.3) to draw the left Kirchhoff graph in Figure 4, or an equivalent version of this graph. Of course there were many free choices in the above construction where one could have added new vertices and/or edges in alternate ways. One set of alternate choices would lead to the other graph in Figure 4. Still other choices could lead to a diverging construction of a larger and larger graph which never satisfies both of the Kirchhoff conditions at all of its vertices and for all of its cycles. A full algorithm for a construction would need a measure of how close/far the process is from converging to determine whether to proceed with an ongoing construction, or to go back to some earlier point in the construction and make a different choice for rows to add or cycles to weave. 4.2 Construction of Kirchhoff Graph in Figure 7 Now we turn our attention to the construction of a Kirchhoff graph for a somewhat larger matrix—A defined by (3.3). To begin, consider vectors u1 and u2 from Null(A) in (3.4) and vectors v1, v2 and v3 from Row(A) in (3.5). These cycles/cuts do not involve either of the overall reactions b1 or b2 and therefore constitute an important subgraph to the Kirchhoff graph we seek. Because the third element of u1 is -1, let us start with row vector v1, and take this as the first row of a partial incidence matrix. Since this is in fact the only row space vector with a nonzero third entry, the only choice for the second row of the incidence matrix is the negative of the first row, and the -2 and 2 in the third column of these first two rows of a partial incidence matrix represent s3 from u1. The next row in the incidence must be connected to the second row by either s5 or s8 (the other two edge vectors in u1). Since the only nonzero entries in the second incidence matrix row are in the first and fifth columns, one should hope to use s5 to connect the second and third rows of the incidence matrix. This third row cannot be a multiple of the first or second since then s3 and s5 would need to be multiples of each other. So the only possible choice among the given row space vectors that s5 can connect to is the negative of row vector v2 in (3.5). To complete this first cycle and represent the rest of the entries v1, the fourth row on the incidence matrix can be the negative of the third. Two copies of s8 then connect the 2 and - 2 in the eight columns of the third and fourth rows of the incidence matrix, and finally an s5 then connects the 1 and -1 in the fifth column of the fourth and first rows, thus completing the cycle and weaving v1, through several row vectors from (3.5). This weave is shown in (4.4) where the bold face numbers correspond to vertices that are in the first cycle. In (4.4), note that each row corresponds to a vertex, and each column corresponds to an edge vector. Since there are two s5 edge vectors in this cycle, the entries corresponding to each vector are realigned to the left or right to indicate which vertices are connected: 1 0 —2 0 — 1 0 0 0 0 0 1 0 2 0 1 0 0 0 0 0 0 0 0 0 — 1 0 1 2 0 0 0 0 0 0 1 0 1 2 0 0 (4.4) Again, the convention is that edge vectors go from positive entries to negative ones, and these entries must always have the same absolute value. Also the non-bold elements of (4.4) represent edge vectors that are currently open (not connected) and must be connected in the eventual full incidence matrix. Now we must try to incorporate the second cycle (i.e., the second vector from (3.4)) into the partial incidence matrix. Although there is no guarantee, one can hope that the s8 edge vector from the first cycle is common to this second cycle and therefore start with it. Because the (3,7) element of (4.4) is 1, it makes sense to try to have s7 leave v3. This edge cannot go to v4 since it would then connect the same vertices as s8. Since one also needs s6 in this cycle, it makes sense to add v3 to the partial incidence matrix, and thereby to add a new vertex to the graph. Adding -v3 as the sixth row of the partial incidence matrix, one connects these rows and their vertices using edge vectors s6 and s7 to complete the second cycle. The rows of the partial incidence matrix for the first six vertices are now given in (4.5); the first cycle is now in italics, while the new cycle is in bold: 1 0 — 2 0 — 1 0 0 0 0 0 1 0 2 0 1 0 0 0 0 0 0 0 0 0 — 1 0 1 2 0 0 0 0 0 0 1 0 —1 —2 0 0 0 0 0 1 0 2 —1 0 0 0 0 0 0 1 0 2 1 0 0 0 Next one must include u3 and u4, the latter two null space vectors from (3.4), and do this in such a way so as to tie up the non-boldface, non-italicized entries in (4.5). Considering u3, one must have two copies of the si edge vector and one s3 in this cycle. The first two vertices (the first two rows of (4.5)) are already connected by two copies of s3 and have loose ends for si in the proper direction, so this would seem a natural place to tie in the third cycle. To complete this cycle, one must add a vertex corresponding to v4 and add an s1 edge vector to connect this seventh row of the partial incidence matrix to its second row. Continuing this cycle, one must add an eighth row to the partial incidence matrix which is two times v5, and two copies of ò1 to connect the vertices that correspond to these seventh and eighth rows. Finally two copies of s2 are needed to complete the cycle, but upon checking each of the rows in (3.5), one finds that none have the correct requirements for a final vertex. This means that linear combinations of the rows in (3.5) must be considered. Taking into account all of the requirements for this final vertex, one finds that the correct linear combination is v4 — 2v5. This linear combination then becomes the ninth row of the partial incidence matrix, and the corresponding vertex is connected to v1 by one copy of s1, and to v8 by two copies of s2, all with the proper orientation to complete the third cycle. The partial incidence matrix for the first nine vertices is thus 1 0 -2 0 -1 0 0 0 0 0 -1 0 2 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 2 0 0 0 0 0 0 1 0 -1 -2 0 0 0 0 0 1 0 2 -1 0 0 0 0 0 0 -1 0 -2 1 0 0 0 1 0 0 0 0 0 0 0 2 1 0 -2 0 0 0 0 0 0 -2 0 1 2 0 0 0 0 0 0 0 1 (4.6) Again the new cycle in (4.6) is in bold, while the two previous cycles are now in italics. In this case, the final cycle corresponding to u4 is easy to include, but it must occur twice. The partial incidence matrix (4.6) has only four loose connections (nonzero entries that are neither in italics or bold), and happily these can be connected pairwise by copies of s4 and b2 through the addition of two new vertices, one corresponding to v6, and the other corresponding to its negative. The full incidence matrix is then 1 0 -2 0 -1 0 0 0 0 0 -1 0 2 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 2 0 0 0 0 0 0 1 0 -1 -2 0 0 0 0 0 1 0 2 -1 0 0 0 0 0 0 -1 0 -2 1 0 0 0 1 0 0 0 0 0 0 0 2 1 0 -2 0 0 0 0 0 0 -2 0 -1 2 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 -1 0 0 0 1 0 0 0 0 0 1 (4.7) For simplicity, no entries in (4.7) are in bold or italics, but entries corresponding to different copies of the same edge are still shifted to the right or left side of their column. Remark 4.2. Again when the edge vectors that satisfy the definition of Kirchhoff graph are actually drawn, (Definition 2.1), their exact length and direction are not important. One needs only to ensure that a given edge vector sj has the same length and orientation each time it appears in the graph, and to avoid any degeneracies (accidentally placing two distinct vertices in the same position). Because a Kirchhoff graph is actually a two dimensional projection of a much higher dimensional geometric structure, the incidence matrix is sufficient to produce some projection of this structure. For example, if the Kirchhoff graph comes from a reaction network with N species, then the vector structure lies in the stoichiometric space QN. Dividing the process into two parts as was done above, first constructing the part of the graph that does not involve the overall reactions (or some other group of reaction steps), then adding on these remaining steps (edges) as the row and null space vectors require, is often a very effective way of constructing a full Kirchhoff graph. Of course the above construction is not guaranteed to produce a Kirchhoff graph, and in particular there are points in the process where one might choose the wrong row and not be able to neatly complete a given cycle. One would then have to go back and consider different rows or linear combinations of rows from (3.5) to form possible rows in a partial incidence matrix. Fortunately this laborious process should be computerizable, so that a computer can sort through the possible weaves to produce a Kirchhoff graph, assuming that one actually exists. 4.3 Summary of Kirchhoff Graph Construction While the construction process used above contains too many free choices to be called an algorithm, it does have a certain structure which it is useful to summarize. Given a matrix A over Q, the process can be summarized in the following general steps: Step 1: Construct bases with integral elements for both Null(A) and Row(A). Even for moderately large matrices, this construction can be accomplished effectively with any of a number of software packages/applications. Step 2: Refine the bases found in Step 1 to find ones that satisfy minimal total absolute sum norms: If [xi} is an integral basis for one of these two spaces, with xi = (xij ) and xij e Z, refine this basis until J2ij Ixij | minimal. Step 3: Starting with the minimal basis for Row(A), begin to construct a first attempt at an incidence matrix for the first vector in Null(A). The construction may require linear combinations of the row vectors. This first attempt should have column pairs which represent each edge in the cycle corresponding to this first vector in Null(A), but the order of the edges may or may not match what is eventually needed, and there will in general be "loose ends," entries in the matrix which represent one end of an edge for which there is not yet an entry for the other end. As in the earlier examples, it may also be necessary to introduce one or more null vertices (a rows of zeros) to form an incidence matrix consistent with this first null space vector. Step 4: Next, attempt to include the next vector from Null(A) in the growing incidence matrix, using as many of the loose ends as possible, but also adding new rows from Row(A) as needed. Note that each additional row in the incidence matrix corresponds to an additional vertex in the developing Kirchhoff graph. Step 5: If all vectors from Null(A) have been included and all loose ends have been ellim-inated, then the construction of the incidence matrix and the Kirchhoff graph are complete. If there remain vectors from Null(A) to be included, but the number of loose ends is in some sense declining, return to Step 4 to add another null space vector. If the number of loose ends is in some increasing, then return to Step 3 and/or Step 4 to change the order in which the edges appear in one or more cycles, or change the row vectors that determine which edges are incident to each vertex. 5 Conclusion A Kirchhoff graph for a rational matrix is a graph-theoretic representation of the fundamental theorem of linear algebra for that matrix. When the matrix is based on a reaction network, a Kirchhoff graph is a circuit diagram for that reaction network, and it represents the connection between the Kirchhoff laws and the fundamental theorem of linear algebra. Figure 8: An example of a relatively simple matrix with a fairly intricate Kirchhoff graph. Edge vector s42 is the vector sum of s2 and s4 which always occur in sequence. The multiplicity of an edge vector between given vertices is given by hash marks which should be read as Roman numerals. To reduce crowding, not all edge vectors are labeled. Certainly the most interesting open question unresolved in this work is the actual existence of a Kirchhoff graph for any given matrix. While it is not obvious that the conjecture is true, it is clear that there is no guarantee that a Kirchhoff graph for a relatively simple matrix will itself be in any sense simple. Consider the matrix 1 -1 1 0 0 A = 0 1 0 1 0 -4 0 0 1 1 Then Null(A) = Span {[0, 1, 1,-1, 1]T, [1, 0,-1, 0, 4]T} (5.2) and Row(A) is the span of the three rows of A. Therefore a Kirchhoff graph for this matrix must have two linearly independent cycles, and all of its vertices must lie in the row space, but this really does not indicate how intricate a Kirchhoff graph for A is. The author believes that the Kirchhoff graph given in Figure 8 is the simplest one corresponding to A in (5.1). It is difficult to image that anyone could guess how complicated a Kirchhoff graph for A needs to be just looking at A, Null(A) and Row(A). Other matrices that would appear similar to A have much simpler Kirchhoff graphs, but there are also other relatively simple matrices (or relatively simple reaction networks) where to the author's knowledge no Kirchhoff graph has yet been constructed and any possible Kirchhoff graph must be large and intricate. For example, consider the matrix A 1003 1829 -17 0 0 59 1411 1003 (5.3) The rows of A in (5.3) clearly form a basis for Row(A), and no linear combination of these two rows will lead to a simpler set of vertex conditions for any possible Kirchhoff graph. One also finds that Null(A) = Span 31 1 -17 0 0 59 1 -83 (5.4) so any possible Kirchhoff graph for this matrix must be a complicated weave of large cycles based on these null space vectors. Again considering linear combinations of these null space vectors would not simplify the graph. The rows of (5.3) and the vectors of (5.4) satisfy the minimal total absolute sum norm condition discussed in Step 2 above. If the matrix in (5.3) is the stoichiometric matrix of some reaction network, even if one succeeds in constructing a Kirchhoff graph, it would seem to be of little use since it would be far too complicated. But since stoichiometric coefficients are typically 0, ±1 or occasionally ±2, elements like those in (5.3) are not realistic as stoichiometric coefficients. From an applications point of view, the key point is that when the Kirchhoff graph for a reaction network is relatively simple, it can be used to study the reaction network in the same way that a circuit diagram is used to study an electrical network. One can construct equivalent circuits, study how temperature, pH or other changes which reaction pathway is dominant, or compute the rate of an overall reaction from the rates of the individual steps, to name a few. In terms of future work, the process for constructing Kirchhoff graphs needs to be computerized. This will make it possible to find Kirchhoff graphs for matrices and reaction networks where construction by hand is too time consuming to be practical. Finally it seems worth noting that there is a certain aesthetic beauty in the intricacy of the more-complicated Kirchhoff graphs such as the one in Figure 8. 6 Acknowledgments The author wishes to thank Manasi Vartek for originally bringing to his attention the NaCl-NaOH reaction network, and also wishes to thank Ravi Datta and Ilie Fishtik for introducing him to the concept of reaction route graphs, and Pete Christopher, Willy Hereman, Bill Martin, Brigitte Servatius, Gil Strang and Lubos Thoma for many illuminating discussions on graph theory and linear algebra. Finally, the author wishes to thank the referees for numerous helpful suggestions and corrections. References [1] B. Bollobas, Modern Graph Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, NY, 1998. [2] T. Bommaraju, P. Orosz and E. 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