ARS MATHEMATICA CONTEMPORANEA Volume 6, Number 1, Spring/Summer 2013, Pages 1-185 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 Society of Mathematicians, Physicists and Astronomers of Slovenia The Institute of Mathematics, Physics and Mechanics The University of Primorska ARS MATHEMATICA CONTEMPORANEA Bled'11 This issue of Ars Mathematica Contemporanea contains a selection of articles presented at the 7th Slovenian Graph Theory Conference (Bled'11), held from June 19 to June 25,2011, by tradition at Lake Bled, Slovenia. This conference (held every four years) has progressed a long way since the first one in 1991. The number of participants has grown, from just 30 in 1991, to 270 at Bled'11, representing 40 countries and all six continents. The Bled'11 conference was attended by some of the leading researchers in graph theory, as well as many postdocs and talented PhD students. There were nine keynote speakers (Jonathan L. Gross, Wilfried Imrich, Alexander A. Ivanov, Läszlö Loväsz, Jaroslav Nešetril, Egon Schulte, Tamäs Szonyi, Martin Škoviera, and Asia Ivic Weiss), plus 213 contributed talks presented within 16 minisymposia, and a general session. The minisymposia brought together researchers from specific fields, ranging across algebraic, algorithmic, geometric, topological, and other aspects of graph theory, and enabled them to present their work and exchange ideas. We know that this led to progress on many open problems, and catalysed many new collaborations. The algebraic minisymposia were dedicated to Henry H. Glover, a very dear and strong collaborator of the Slovenian Algebraic Graph Theory group, who sadly passed away just a few weeks before the conference. Many special and satellite events were organised during and after the conference. We celebrated the 70th birthdays of Jonathan L. Gross and Wilfried Imrich. A meeting of the International Academy of Mathematical Chemistry was held, as well as the first meeting of the team from the ESF EuroGiGA GReGAS research project, led by Tomaž Pisanski (University of Primorska). Another highlight was the 'Milestones' exhibition, due to Boštjan Kuzman, which presented many important steps in the development of graph theory in Slovenia — from the first lecture notes, scientific results, published papers and doctoral theses, to international collaborations, celebrated publications, editorial positions, establishments of new institutions, scientific journals, and further projects. Immediately after the conference, about 80 participants attended a satellite PhD Summer School — the Algebraic Graph Theory Summer School, held in Rogla, and organised by the University of Primorska, and some participants attended a 'Mathematics meets Art' event in Ljubljana. In the past, many papers from each Bled conference were published in a special issue of the Discrete Mathematics journal. In 2007 however, when Ars Mathematica Contemporanea was established, the participants of the 6th Bled conference were given the option of publishing their contributions in a special issue of this new journal. The organisers of the 2011 conference decided that the contributions related to Bled'11 would be published exclusively by Ars Mathematica Contemporanea. This is the first such special issue, and contains 15 articles, accepted for publication after a thorough refereeing process. In producing it, we are able to present to the readers of this journal a selected number of the Bled'11 conference contributions, containing high quality results, and establishing starting points for future research. More of these will be published in another special issue in 2014. Klavdija Kutnar and Primož Šparl Guest Editors ARS MATHEMATICA CONTEMPORANEA Contents On regular and equivelar Leonardo polyhedra Gäbor Gévay, Jörg M. Wills ......................... 1 Line graphs and geodesic transitivity Alice Devillers, Wei Jin, Cai Heng Li, Cheryl E. Praeger.......... 13 Johnson graphs are Hamilton-connected Brian Alspach.................................21 Nonorientable regular maps over linear fractional groups Gareth A. Jones, Martin MaCaj, Jozef ŠirM.................25 Embeddings of cubic Halin graphs: Genus distributions Jonathan L. Gross............................... 37 Markov chain algorithms for generating sets uniformly at random Alberto Policriti, Alexandru I. Tomescu................... 57 Complex parameterization of triangulations on oriented maps Mathieu Dutour Sikiric............................69 On the minimum rainbow subgraph number of a graph Ingo Schiermeyer............................... 83 A note on domination and independence-domination numbers of graphs Martin Milanic................................ 89 On D. G. Higman's note on regular 3-graphs Daniel Kalmanovich .............................99 Some properties of the Zagreb eccentricity indices Kinkar Ch. Das, Dae-Won Lee, Ante Graovac................117 Consensus strategies for signed profiles on graphs Kannan Balakrishnan, Manoj Changat, Henry Martyn Mulder, Ajitha R. Subhamathi.............................127 Quasi m-Cayley circulants Ademir Hujdurovic..............................147 2-Groups that factorise as products of cyclic groups, and regular embed-dings of complete bipartite graphs Shaofei Du, Gareth Jones, Jin Ho Kwak, Roman Nedela, Martin Škoviera . 155 Sharp spectral inequalities for connected bipartite graphs with maximal Q-index Milica Andelic, C. M. da Fonseca, Tamara Koledin, Zoran Stanic......171 Volume 6, Number 1, Spring/Summer 2013, Pages 1-185 d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 1-11 On regular and equivelar Leonardo polyhedra Gabor Gevay Bolyai Institute, University of Szeged, Aradi vértanuk tere 1, H-6720 Szeged, Hungary Jorg M. Wills Math. Institute, University of Siegen, Emmy-Noether-Campus, D-57068 Siegen, Germany Received 12 July 2011, accepted 4 January 2012, published online 1 June 2012 Abstract A Leonardo polyhedron is a 2-manifold without boundary, embedded in Euclidean 3-space E3, built up of convex polygons and with the geometric symmetry (or rotation) group of a Platonic solid and of genus g > 2. The polyhedra are named in honour of Leonardo's famous illustrations in [19] (cf. also [12]). Only six combinatorially regular Leonardo polyhedra are known: Coxeter's four regular skew polyhedra, and the polyhedral realizations of the regular maps by Klein of genus 3 and by Fricke and Klein of genus 5. In this paper we construct infinite series of equivelar (i.e. locally regular) Leonardo polyhedra, which share some properties with the regular ones, namely the same Schläfli symbols and related topological structure. So the weaker condition of local regularity allows a much greater variety of symmetric polyhedra. Keywords: Equivelar polyhedron, Leonardo polyhedron, regular polyhedron, genus, Schläfli symbol, symmetry group. Math. Subj. Class.: 52B15, 52B70 1 Introduction A polyhedron is a compact 2-manifold without boundary embedded in Euclidean 3-space E3, hence oriented. It is built up of finitely many (planar) convex polygons, any two of which meet, if at all, in a single edge or a single vertex. If v, e and f denote the number of vertices, edges and faces, respectively, of the polyhedron, then one has the basic Euler-Poincare equation v - e + f = 2 - 2g = X, E-mail addresses: gevay@math.u-szeged.hu (Gabor Gévay), wills@mathematik.uni-siegen.de (Jorg M. Wills) where g > 0 denotes the genus and x the Euler characteristic. In this paper we do not consider the case of tori (g = 1), but only the polyhedra with g > 2. If all faces of a polyhedron arep-gons, p > 3, and all vertices q-valent, q > 3, then the polyhedron is called locally regular or equivelar and is denoted by its Schlafli symbol {p, q} (cf. [3, 15]). We note that the extended Schlafli symbol {p, q; g} is also used. A much stronger condition is (global combinatorial) regularity: a polyhedron is called regular if its automorphism group acts transitively on its flags (incidence triples of vertex, edge and face). Regular maps and their groups play a central role in classical complex analysis and algebraic geometry (e.g. Riemann surfaces, automorphic functions, Poincaré model). Hence regular polyhedra can be interpreted as 3D geometric models or visualizations of regular maps, and they are closely related to the Platonic solids. The geometric (or Euclidean) symmetry group of the polyhedron is the group of isometries of E3 stabilizing the polyhedron. It is a subgroup of the automorphism group; to be precise, the automorphism group has a (proper, or improper) subgroup that is isomorphic to the geometric symmetry group. For a combinatorially regular polyhedron the geometric symmetry group is, in general, much smaller than the automorphism group; they coincide only in the case of Platonic solids. For any polyhedron with given combinatorial structure we tacitly assume that it has maximal geometric symmetry. Polyhedra which have the geometric rotation or full symmetry group of a Platonic solid deserve particular interest. They are called Leonardo polyhedra, because Leonardo was the first to draw such polyhedra in Luca Pacioli's book [19] in 1500-1503 (see also [6] and [12]). It is easy to check that the polyhedra in this book are neither equivelar nor regular. Leonardo also drew some polyhedra with lower symmetry groups (e.g. dihedral), but we only use the name for Platonic symmetries. Obviously there are no Leonardo polyhedra of genus g = 1, because tori can have at most dihedral symmetry. For similar reasons there are no Leonardo polyhedra with g = 2. For g = 3 there are some with tetrahedral symmetry. 2 Regular and equivelar Leonardo polyhedra Regular Leonardo polyhedra seem to be very rare. Only six are known yet. The first four are Coxeter's regular skew polyhedra [5], first discovered by Coxeter in 1937 and partially by Alicia Boole Stott already in 1913 [1]. There is one dual pair of genus g = 6, with tetrahedral symmetry and of type {4,6} and {6,4}, and one dual pair of genus g = 73, with octahedral symmetry and of type {4,8} and {8,4}. (In standard notation: {4, 6|3}, {6,4|3}, {4, 813} and {8,4|3}, cf. [5, 15, 21].) The spines of Coxeter's regular skew polyhedra are isomorphic to the 1-skeletons of the regular 4-simplex or the regular 24-cell, i.e. the only self-dual regular 4-polytopes. We note that the term spine, borrowed from topology, is meant here as a graph, embedded in E3, such that its regular neighbourhood [18] is a 3-manifold with boundary, and the boundary of this manifold is just our polyhedron. Furthermore, there is the polyhedral realization [20] of Felix Klein's regular map of genus 3 with tetrahedral rotation group and of type {3, 7}. Its dual with non-convex heptagons was recently discovered [13], but here we only consider polyhedra with convex faces. The sixth regular polyhedron is the realization of the regular map of Fricke and Klein from 1890. This polyhedron was found by Griinbaum and Shephard in 1984 [11], because of its vertex-transitivity. But its regularity was only recently discovered [2]. It is of type {3, 8}, with genus 5, and it has octahedral rotation symmetry. The spines of these last two polyhedra are isomorphic to the 1-skeleton of the tetrahedron or the cube, hence of convex 3-polytopes in both cases. No other regular Leonardo polyhedra are known yet. In this paper we construct series of equivelar polyhedra, which are related to the previous 6 regular polyhedra: Theorem 2.1. There are infinite series of equivelar Leonardo polyhedra with tetrahedral, octahedral and dodecahedral symmetry group and of Schlafli type {3,7}, {3, 8}, {3,9}, {4, 6} and {6,4}, and whose spine is isomorphic to the 1-skeleton of a convex 3- or 4-polytope. Remark 2.2. The result shows that there are infinite series of equivelar polyhedra, which are closely related to the regular Leonardo polyhedra. Only the types {4,8} and {8,4} are missing. Remark 2.3. In the previous papers [9] and [23] the authors provided infinite series of equivelar Leonardo polyhedra of type {4, 6} and {6,4}. But these were of very different spatial structure than the six known regular ones, as they are built up of connected shells (like an onion). The search for closer equivelar analogues was one motivation for this paper. The other one was the recently discovered regularity [2] of the Griinbaum-Shephard polyhedron. Among the polyhedra of "small" Schlafli-type (i.e. those with p + q < 12) the equivelar polyhedra of type {4, 5} and {5,4} differ from the others, as follows. The only known regular polyhedra of these types have genus 5 and a small symmetry group of order 4, namely (Z2 x Z2) (cf. [17] and [14]). The equivelar Leonardo polyhedra of this type also differ from the others as the following result shows. Theorem 2.4. a) Type {4, 5}: There exist four infinite series of equivelar Leonardo polyhedra with the following genera and symmetry groups: • g = 1 + 6k with tetrahedral symmetry group, in two non-isomorphic versions; • g = 1 + 12k with octahedral symmetry group; • g = 1 + 30k with icosahedral symmetry group (k = 1,2,... ). b) Type {5,4}: There exist equivelar Leonardo polyhedra with the following genera and symmetry groups: • g =13, 31 with tetrahedral symmetry group; • g = 7,13, 25, 97, 289 with octahedral symmetry group; • g = 31, 61, 3601 with icosahedral symmetry group. In [22] there is a pair of equivelar Leonardo polyhedra of type {4,5} and {5,4} with g = 7 and octahedral symmetry, and in Figure 1 we show a new example of type {4,5} with g =19 and icosahedral (rotation) symmetry. This polyhedron consists of an outer and an inner shell, homothetic to each other and positioned concentrically. Both of them are composed of 60 quadrangular faces; besides, there are 20 triangular holes in each. The two Figure 1: The Leonardo polyhedron of type {4,5; 19}; f = 36(4,10, 5). shells are joined by triangular prismatic tubes along these holes. Since this polyhedron has shortest non-0-homotopic paths of length 3 and 4, it is not regular. It follows from Conder's list of regular maps [4] that for g =19 there is precisely one regular map of type {4,5}. So the only possible Leonardo polyhedron of such type would be a realization of this map, and it is an open question if it exists or not. We note also that the polyhedron of type {5,4; 7} with octahedral symmetry group was found already in 1983 (see [22], Figure 2). In [9] it is constructed in a slightly different way. The other related types are new and here we show in Figure 2 the example with genus 13. 3 Proof for the existence of equivelar series In this section we prove Theorems 2.1 and 2.4 by constructing the polyhedra in question. Proof of Theorem 2.1 Construction for Schläfli type {3,7}. The construction was already done in [16], but without any symmetry assumptions. We split our proof into three parts. First we show that there are infinitely many simple convex 3-polytopes, i.e. such that all their vertices are 3-valent, for each of the required symmetry groups. The first polytopes of this type are the tetrahedron, the cube and the dodecahedron. They have 4, 8 and 20 3-valent vertices, respectively. We now continue by induction. Figure 2: The Leonardo polyhedron of type {5,4; 13}; f = 24(5,10,4). For a simple convex polytope with v vertices we cut off these vertices by a plane each, such that no cuts intersect and that the global symmetry is preserved. We obtain a new convex 3-polytope with the same symmetry group and with 3v vertices, all of them 3-valent. Clearly this polytope is not equivelar, because it contains faces of different type. The second step is to construct from each of these polytopes a new one with 5-valent vertices. For each of the convex polytopes with 3v vertices we make the following operation. We shrink all faces by the same factor, such that each face remains in its given affine hull. Hence they are disjoint. Now we take the convex hull of the system of these new polygons, so that we obtain a new convex polytope with 4-valent vertices. Each vertex of the former polytope corresponds to a triangle, so does each edge to a quadrangle, and the vertex-number of the new polytope is 9v. We now split the new quadrangles by a diagonal into two triangles in the right order, such that the global symmetry is preserved. More precisely, by this operation the full symmetry group is lost and reduced to the rotation subgroup of index two. We now have a convex 3-polytope with 5-valent vertices and the required symmetry. The third step leads us to the construction of tunnels and the required polyhedron of type {3,7}. First we take the boundary complex of our polytope, and put a smaller copy of this complex into the former one with the same centre and orientation. Then we delete in both objects the shrinked polygons. The remaining faces are all triangles. We now connect any two corresponding holes by tunnels, built up of quadrangles. Again we split each of these quadrangles by a diagonal in the right order, so that the symmetry is preserved. We obtain a polyhedron of Schlafli type {3, 7} with the required symmetry. Finally, by a slight rotation of the holes (originating from the shrinked faces), each within its affine hull and to different extent in the two spherical complexes, it is ensured that no adjacent triangles are coplanar (this can also be done in symmetry-preserving way). Construction for Schlafli types {4,6} and {6,4}. The construction was already described in [16], although without any symmetry considerations. In order to make the paper self-contained, we sketch the proof for the {6,4} series. The dual {4, 6} series are constructed similarly. Let P be one of the simple 3-polytopes with Platonic symmetry group obtained in the first step of our former proof and let SD be the Schlegel diagram of the 4-prism with base P such that it is in one of the Platonic bases of the prism. All vertices of SD are 4-valent. We take the midpoints of all edges of SD, and then the convex hull of the midpoints of any four edges which are incident to a vertex of SD. Thus to each vertex of SD corresponds a 3-simplex and each vertex of a simplex is shared with a vertex of a neighboring simplex. Now we enlarge each simplex by the same factor 1 + e, e > 0 sufficiently small. We delete those parts of the simplices which lie inside another simplex and obtain a polyhedron of type {6,4} with the required symmetry properties. Construction for Schlafli-types {3,8} and {3,9}. From each polyhedron of type {6,4} we obtain one of type {3,8} as follows. In each hexagon one connects a triplet of non-consecutive vertices by segments and obtains a tiling of the hexagon into 4 triangles. If one does this in the right order on the whole {6,4} polyhedron, one obtains the required {3,8} polyhedron. The {3,9} series is obtained from the {4, 6} series as follows. Each quadrangle can be divided into two triangles by a diagonal. If one does this in the right order on the whole polyhedron, one obtains the required {3, 9} polyhedron. The crucial point for this procedure (which was already described in [16]) is the fact that, when applied to any polyhedron of type {p, q}, the valency q of the vertices is even. □ Proof of Theorem 2.4 Construction for Schlafli-type {4,5}. Start from two distinct types of Archimedean polyhedra, Pi and P2. Pi is the truncated octahedron with six square faces and eight hexagonal faces. P2 is the rhombicuboctahe-dron, which has 6+12 square faces and eight triangular faces. For the following construction it is crucial that Pi has only 3-valent vertices, and P2 only 4-valent vertices. Note that both the octuple of the hexagonal faces of Pi and the octuple of the triangular faces of P2 decomposes to two disjoint classes. In each case such a class is a quadruple forming an orbit under the action of the tetrahedral symmetry group (a subgroup of the octahedral symmetry group of these polyhedra). Delete these hexagonal and triangular faces, and denote the complexes obtained in this way by Pi and P2, respectively. Take 2k (k > 2) concentric and homothetic copies of Pi. We call them shells of our polyhedron under construction. Now join the holes of the neighbouring shells by hexagonal prismatic tubes. The tubes are arranged so that each intermediate shell is joined to its outer or inner neighbour, in both cases using four tubes and using holes that belong to the same class (but different in the two cases). The innermost shell is joined to the outermost shell. To avoid undesirable contacts, the holes of the outermost shell are shrinked to a suitable size with respect to those of the intermediate shells, while keeping the symmetry. For P2', the construction is the same with the only difference that here one uses triangular tubes. We obtain two different infinite series of polyhedra, both of the desired Schlafli type, and with the (full) tetrahedral symmetry group and genus g = 1 + 6k (k = 1, 2,... ). Consider again the four orbits of faces of the rhombicuboctahedron with respect to the action of the tetrahedral group. Clearly, analogous polyhedra can be constructed, likewise with four orbits of faces with respect to the octahedral and the icosahedral group. In the octahedral case these orbits are 6 squares, 8 regular triangles, 12 rhombi and 24 rectangles, while in the icosahedral case there are 12 regular pentagons, 20 regular triangles, 30 squares and 60 symmetric trapezia. In both cases all the vertices are 4-valent (and the polar dual is such that all the faces are kite-shaped and form two orbits). Now deleting the square and triangular faces in the the octahedral case, and the non-quadrangular faces in the icosahedral case, and applying an analogous construction as above, one obtains the desired infinite series of polyhedra with genus g = 1 + 12k and g = 1 + 30k, respectively. The starting member of the icosahedral series is shown in Figure 3. Figure 3: The Leonardo polyhedron of type {4,5; 31}; f = 60(4,10, 5). Construction for Schlafli-type {5,4}. Let P be a polyhedron satisfying the following conditions: (1) the symmetry group G(P) of P is equal to the full symmetry group of one of the Platonic solids; (2) G(P) is transitive on the faces of P ; (3) the faces of P are quadrangles; (4) each edge of P is contained in one of the mirror planes determined by G(P). It is easy to see that P is combinatorially equivalent to one of the following five polyhedra (cf. [8]): • cube; • rhombic dodecahedron; • rhombic triacontahedron; • deltoidal icositetrahedron (dual of the rhombicuboctahedron) (see e.g. Figure 3 in [9]; • deltoidal hexecontahedron (dual of the Archimedean polyhedron called rhombicosi-dodecahedron) (see e.g. Figure 5 in [9] or Figure 9 in [7]). Put on each of the quadrangular faces of P a bipyramid, each pairwise congruent, such that the midpoints of the edges of the face form the basal vertices of the bipyramid. Then enlarge each bipyramid from the centre of its own base by the same factor 1 + e, e > 0 sufficiently small. Delete now those parts of the bipyramids which lie inside another bipyramid (along with the original faces of P). One obtains a polyhedron of the desired Schlafli type such that its symmetry group remains the same as that of the example of P we started from. The genus of this polyhedron is g = f2(P ) + 1, where f2(P ) is the number of the faces of P ; hence the genera in the five cases above are 7,13,31,25 and 61, respectively. Finally, we note that there is a polyhedron called deltoidal dodecahedron, combinatorially equivalent to the rhombic dodecahedron but with tetrahedral symmetry (a well-known figure in geometric crystallography, see e.g. [10]). It also satisfies the conditions above. Hence starting from it, our construction provides the tetrahedrally symmetric polyhedron with g =13. The conditions (1-4) above can be suitably modified such that they are satisfied by equivelar polyhedra with quadrangular faces and with spine isomorphic to the 1-skeleton of a regular 4-polytope Q. (Two of these polyhedra are even regular, namely that of type {4, 6; 6} and {4,8; 73} [21].) Thus, performing the construction in E4, then taking a suitable projection to E3, one obtains Leonardo polyhedra of the following genera: g = 31 (Q is the regular 4-simplex), g = 97 (Q is either the 4-cube or the regular 16-cell), g = 289 (Q is the regular 24-cell) and g = 3601 (Q is either the regular 120-cell or the 600-cell). □ In conclusion, we present a sporadic example of a polyhedron of type {3,8; 7}, which differs in its structure from those in Theorem 2.1. It is constructed from two solids P and P ', such that P ' is a non-convex version of the convex 3-polytope P ; their boundary are combinatorially equivalent to each other, and have the f-vector f = (30, 84, 56). The size and shape of the bounding polyhedra is adjusted so that deleting a whole 8-element orbit of faces from both, the complexes obtained in this way can be glued together along the holes, thus forming the outer and inner shell of a new polyhedron. This polyhedron has octahedral rotation symmetry; it is shown in Figure 4a, and its inner shell in Figure 4b. (b) The inner shell of the polyhedron. Figure 4: The Leonardo polyhedron of type {3, 8; 7}; f = 12(3,12,8). References [1] A. Boole Stott, Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam 11 (1913), 3-24. [2] U. Brehm, B. Grünbaum and J. M. Wills, Polyhedral realization of the regular Fricke-Klein map of genus 5, in preparation. [3] U. Brehm and J. M. Wills, Polyhedral manifolds, in: P. M. Gruber and J. M. Wills (eds.), Handbook of Convex Geometry, North-Holland, Amsterdam, 1993, 535-554. [4] http://www.math.auckland.ac.nz/~conder/ [5] H. S. M. Coxeter, Regular skew polyhedra in three and four dimensions and their topological analogues, Proc. London Math. Soc. 43 (1937), 33-62. Reprinted in: H. S. M. Coxeter, Twelve Geometric Essays, Southern Illinois University Press, Carbondale, 1968, 75-105. [6] J. V. Field, Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Durer, Daniele Barbaro, and Johannes Kepler, Arch. Hist. Exact Sci. 50 (1997), 241-289. [7] G. Gevay, Icosahedral morphology, in: I. Hargittai (ed.), Fivefold Symmetry, World Scientific, Singapore, 1992, 177-203. [8] G. Gevay, Kepler hypersolids, in: K. Boroczky and G. Fejes Toth (eds.), Intuitive Geometry, North-Holland, Amsterdam - Math. Soc. Janos Bolyai, Budapest, 1994, 119-129. [9] G. Gevay, Constructions for some classes of equivelar skew polyhedra, Symmetry Cult. Sci. 22, No. 3-4(2011), 327-345. [10] G. Gevay and K. Miyazaki, Some examples of semi-nodal perfect 4-polytopes, Publ. Math. Debrecen 63-4 (2003), 715-735. [11] B. Grunbaum and G. C. Shephard, Polyhedra with transitivity properties, C. R. Math. Rep. Acad. Sci. Canada 6 (1984), 61-66. [12] D. Huylebrouk, Lost in triangulation: Leonardo da Vinci's mathematical slip-up, Scientific American, March 29, 2011. [13] D. I. McCooey, A non-self-intersecting polyhedral realization of the all-heptagon Klein map, Symmetry Cult. Sci. 20 (2009), 247-268. [14] P. McMullen, E. Schulte and J. M. Wills, Infinite series of combinatorially regular polyhedra in three-space, Geom. Dedicata 26 (1988), 299-307. [15] P. McMullen and E. Schulte, Abstract Regular Polytopes, Cambridge University Press, Cambridge, 2002. [16] P. McMullen, C. Schulz and J. M. Wills, Equivelar polyhedral manifolds in E3, Israel J. Math. 41 (1982), 331-346. [17] P. McMullen, C. Schulz and J. M. Wills, Polyhedral 2-manifolds in E3 with unusually large genus, Israel J. Math. 46 (1983), 127-144. [18] E. E. Moise, Geometric Topology in Dimensions 2 and 3, Springer-Verlag, New York, 1977. [19] L. Pacioli, De Divina Proportione (Disegni di Leonardo da Vinci 1500-1503), Faksimile Do-miniani, Como, 1967. [20] E. Schulte and J. M. Wills, A polyhedral realization of Felix Klein's map {3, 7}8 on a Riemann surface of genus 3, J. London Math. Soc. 32 (1985), 539-547. [21] E. Schulte and J. M. Wills, On Coxeter's regular skew polyhedra, Discrete Math. 60 (1986), 253-262. [22] J. M. Wills, Semi-Platonic manifolds, in: P. M. Gruber and Jörg M. Wills (eds.), Convexity and its Applications, Birkhauser, Basel, 1983, 413-421. [23] J. M. Wills, Combinatorially regular Leonardo polyhedra, Symmetry Cult. Sci. 22, No. 1-2 (2011), 55-64. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 13-20 Line graphs and geodesic transitivity* Alice Devillers , Wei Jin, Cai Heng Li, Cheryl E. Praeger Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia Received 21 October 2011, accepted 28 March 2012, published online 1 June 2012 Abstract For a graph r, a positive integer s and a subgroup G < Aut(T), we prove that G is transitive on the set of s-arcs of r if and only if r has girth at least 2(s - 1) and G is transitive on the set of (s - 1)-geodesics of its line graph. As applications, we first classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive. Secondly we prove that the only non-complete locally cyclic 2-geodesic transitive graphs are the octahedron and the icosahedron. Keywords: Line graphs, s-geodesic transitive graphs, s-arc transitive graphs. Math. Subj. Class.: 05E18, 20B25 1 Introduction A geodesic from a vertex u to a vertex v in a graph is a path of shortest length from u to v. In the infinite setting geodesics play an important role, for example, in the classification of infinite distance transitive graphs [11], and in studying locally finite graphs, see for example, [17]. They are also used to model, in a finite network, the notion of visibility in Euclidean space [22]. Here we study transitivity properties on geodesics in finite graphs. Throughout this paper, we assume that all graphs are finite simple and undirected. Let r be a connected graph with vertex set V(r), edge set E(r) and automorphism group Aut(r). For a positive integer s, an s-arc of r is an (s + 1)-tuple (v0, vi,..., vs) of vertices such that vž, vi+1 are adjacent and vj_1 = vj+1 for 0 < i < s - 1,1 < j < s - 1; it is an s-geodesic if in addition v0, vs are at distance s. For G < Aut(r), r is said to be (G, s)-arc transitive or (G, s)-geodesic transitive, if r contains an s-arc or s-geodesic, * This paper forms part of Australian Research Council Federation Fellowship FF0776186 held by the fourth author. The first author is supported by UWA as part of the Federation Fellowship project during most of the work for this paper. The second author is supported by the Scholarships for International Research Fees (SIRF) at UWA. E-mail addresses: alice.devillers@uwa.edu.au (Alice Devillers), 20535692@student.uwa.edu.au (Wei Jin), cai.heng.li@uwa.edu.au (Cai Heng Li), cheryl.praeger@uwa.edu.au (Cheryl E. Praeger) and G is transitive on the set of t-arcs or t-geodesics respectively for all t < s. Moreover, if G = Aut(r), then G is usually omitted in the previous notation. The study of (G, s)-arc transitive graphs goes back to Tutte's papers [18, 19] which showed that if r is a (G, s)-arc transitive cubic graph then s < 5. About twenty years later, relying on the classification of finite simple groups, Weiss [21] proved that there are no (G, 8)-arc transitive graphs with valency at least three. The family of s-arc transitive graphs has been studied extensively, see [2, 9, 15, 16, 20]. Here we consider these properties for line graphs. The line graph L(r) of a graph r is the graph whose vertices are the edges of r, with two edges adjacent in L(r) if they have a vertex in common. Our first aim in the paper is to investigate connections between the s-arc transitivity of a connected graph r and the (s - 1)-geodesic transitivity of its line graph L(r) where s > 2. A key ingredient in this study is a collection of injective maps Ls, where Ls maps the s-arcs of r to certain s-tuples of edges of r (vertices of L(r)) as defined in Definition 2.3. The major properties of Ls are derived in Theorem 2.4 and the main consequence linking the symmetry of r and L(r) is given in Theorem 1.1, which is proved in Subsection 2.2. We denote by r(u) the set of vertices adjacent to the vertex u in r. If |r(u) | is independent of u G V(r), then r is said to be regular. The girth of r is the length of the shortest cycle; the diameter diam(r) of r is the maximum distance between two vertices. Theorem 1.1. Let r be a connected regular, non-complete graph of girth g and valency at least 3. Let G < Aut(r) and let s bea positive integer such that 2 < s < diam(L(r)) +1. Then G is transitive on the set of s-arcs of r if and only if s < g/2 + 1 and G is transitive on the set of (s — 1)-geodesics of L(r). It follows from a deep theorem of Richard Weiss in [21] that if r is a connected s-arc transitive graph of valency at least 3, then s < 7. This observation yields the following corollary, and its proof can be found in Subsection 2.2. Corollary 1.2. Let r and g be as in Theorem 1.1 . Let s be a positive integer such that 2 < s < diam(L(r)) + 1. If L(r) is (s — 1)-geodesic transitive, then either 2 < s < 7 or s > max{7, g/2 + 1}. Note that in a graph, 1-arcs and 1-geodesics are the same, and are called arcs. For graphs of girth at least 4, each 2-arc is a 2-geodesic so the sets of 2-arc transitive graphs and 2-geodesic transitive graphs are the same. However, there are also 2-geodesic transitive graphs of girth 3. For such a graph r, the subgraph [r(u)] induced on the set r(u) is vertex transitive and contains edges. Moreover, if [T(u)] is complete, then so is r. A vertex transitive, non-complete, non-empty graph must have at least 4 vertices and thus valency 4 is the first interesting case. As an application of Theorem 1.1, we characterise connected non-complete 2-geodesic transitive graphs of girth 3 and valency 4. In this case, [T(u)] = C4 or 2K2 for each u G V(r). If r is s-geodesic transitive with s = diam(r), then r is called geodesic transitive. A graph r is said to be distance transitive if its automorphism group is transitive on the ordered pairs of vertices at any given distance. Theorem 1.3. Let r be a connected non-complete graph of girth 3 and valency 4. Then V is 2-geodesic transitive if and only if r is either L(K4) = O or L(E) for a connected 3-arc transitive cubic graph E. Moreover, r is geodesic transitive if and only if r = L(E) for a cubic distance transitive graph E, namely E = K4, K3,3, the Petersen graph, the Heawood graph or Tutte's 8-cage. Since there are infinitely many 3-arc transitive cubic graphs, there are therefore infinitely many 2-geodesic transitive graphs with girth 3 and valency 4. Theorem 1.3 is proved in Section 3, and it provides a useful method for constructing 2-geodesic transitive graphs of girth 3 and valency 4 which are not geodesic transitive, an example being the line graph of a triple cover of Tutte's 8-cage constructed in [14]. The line graphs mentioned in the second part of Theorem 1.3 are precisely the distance transitive graphs of valency 4 and girth 3 given, for example, in [4, Theorem 7.5.3 (i)]. A graph r is said to be locally cyclic if [r(u)] is a cycle for every vertex u. In particular, the girth of a locally cyclic graph is 3. It was shown in [8, Theorem 1.1] that for 2-geodesic transitive graphs r of girth 3, the subgraph [r(u)] is either a connected graph of diameter 2, or isomorphic to the disjoint union mKr of m copies of a complete graph Kr with m > 2,r > 2. Thus one consequence of Theorem 1.3 is a classification of connected, locally cyclic, 2-geodesic transitive graphs in Corollary 1.4: for [r(u)] = Cn has diameter 2 only for valencies n = 4 or 5, and the valency 5, girth 3, 2-geodesic transitive graphs were classified in [7]. Its proof can be found at the end of Section 3. We note that locally cyclic graphs are important for studying embeddings of graphs in surfaces, see for example [10, 12, 13]. Corollary 1.4. Let r be a connected, non-complete, locally cyclic graph. Then r is 2-geodesic transitive if and only if r is the octahedron or the icosahedron. 2 Line graphs We begin by citing a well-known result about line graphs. Lemma 2.1. [1, p.1455] Let r be a connected graph. If r has at least 5 vertices, then Aut(r) e Aut(L(r)). The subdivision graph S(r) of a graph r is the graph with vertex set V (r) U E (r) and edge set {{u, e}|u G V (r), e G E (r), u G e}. The link between the diameters of r and S(r) was determined in [6, Remark 3.1 (b)]: diam(S(r)) = 2diam(r) + S for some S g {0,1,2}. Here, based on this result, we will show the connection between the diameters of r and L(r) in the following lemma. Lemma 2.2. Let r be a connected graph with |V(r)| > 2. Then it holds diam(L(r)) = diam(r) + x for some x G {-1,0,1}. Moreover, all three values occur, for example, if r = K3+x, then diam(L(r)) = diam(r) + x = 1 + x for each x. Proof. Let d = diam(r), dl = diam(L(r)) and ds = diam(S(r)). Let (x0, x2,..., x2dl ) be a dl-geodesic of L(r). Then by definition of L(r), each edge intersection x2i n x2i+2 is a vertex v2i+1 of r and (x0, v1, x2,..., v2dl-1, x2dl ) is a 2dl-path in S(r). Suppose that (x0, v1, x2,..., v2dl_1, x2dl ) is not a 2dl-geodesic of S(r). Then there is an r-geodesic from xo to x2dl, say (yo, Vi,V2, ■■■,Vr ) with yo = xo and yr = x2dl, such that r < 2di. Since both x0, x2dl are in V(L(r)), it follows that r is even, and hence dL(r) (x0, x2dl ) = 2 < dl which contradicts the fact that (x0, x2,..., x2dl ) is a dl-geodesic of L(r). Thus (x0, vi, x2,..., v2di-1, x2di ) is a 2dl-geodesic in S(r). It follows from [6, Remark 3.1 (b)] that di < ds/2 < d + 1. Now take a ds-geodesic (x0, x1,..., xds ) in S(r). If x0 e E(r), then (x0, x2, x4,..., x2\ds/2\ ) is a |_ds/2j-geodesic in L(r), so di > |_ds/2J > d. Similarly we see that di > d if xds e E(r). Finally if both x0,xds e V(r), then ds is even and dr(x0,xds ) = ds/2. Moreover (x1, x3,..., xds-1 ) is a ()-geodesic in L(r). By [6, Remark 3.1 (b)], ds = 2d, so di > 'ds-. = d — 1. □ 2.1 The map Ls Let r be a finite connected graph. For each integer s > 2, we define a map from the set of s-arcs of r to the set of s-tuples of V(L(r)). Definition 2.3. Let a = (v0, v1,..., vs) be an s-arc of r where s > 2, and for 0 < i < s, let ei := {vi,vm} e E(r). Define Ls(a) := (eo, e1,..., es-1). The following theorem gives some important properties of Ls. Theorem 2.4. Let s > 2, let r be a connected graph containing at least one s-arc, and let Ls be as in Definition 2.3. Then the following statements hold. (1) Ls is an injective map from the set of s-arcs of r to the set of (s — 1)-arcs of L(r). Further, Ls is a bijection if and only if either s = 2, or s > 3 and V = Cm or Pn for some m > 3,n > s. (2) Ls maps s-geodesics of r to (s — 1)-geodesics of L(r). (3) If s < diam(L(r)) +1, then the image Im(Ls ) contains the set Gs-1 of all (s — 1)-geodesics of L(r). Moreover, Im(Ls) = Gs-1 if and only if girth(r) > 2s — 2. (4) Ls is Aut(r)-equivariant, that is, Cs(a)9 = Cs(a9) for all g e Aut(r) and all s-arcs a of r. Proof. (1) Let a = (v0, v1,..., vs) be an s-arc of r and let Ls(a) := (e0, e1,..., es-1) with the ei as in Definition 2.3. Then each of the ei lies in E(r) = V(L(r)) and ek = ek+1 for 0 < k < s — 2. Further, since v j = vj+1, vj+2 for 1 < j < s — 2, we have ej-1 = ej+1. Thus Ls(a) is an (s — 1)-arc of L(r). Let b = (u0, u1,..., us) and c = (w0, w1,..., ws) be two s-arcs of r. Then Ls(b) = (fo,f1,. ..,fs-1) and Ls(c) = (go,g1,... ,gs-1) are two (s — 1)-arcs of L(r), where fi = {ui,ui+1} and gi = {wi,wi+1} for 0 < i < s. Suppose that Ls(b) = Ls(c). Then fi = gi for each i > 0, and hence fi n fi+1 = gi n gi+1, that is, ui+1 = wi+1 for each 0 < i < s — 2. So also u0 = w0 and us = ws, and hence b = c. Thus Ls is injective. Now we prove the second part. Each arc of L(r) is of the form h = (e, f ) where e = {u0, u1} and f = {u1, u2} are distinct edges of r. Thus u0 = u2, so k = (u0, u1, u2) is a 2-arc of r and L2 (k) = h. It follows that L2 is onto and hence is a bijection. If s > 3 and r = Cm or Pn for some m > 3, n > s, then L(r) = Cm or Pn-1 respectively, and hence for every (s — 1)-arc x of L(r), we can find an s-arc y of r such that Ls (y) = x, that is, Ls is onto. Thus Ls is a bijection. Conversely, suppose that Ls is onto, and that s > 3. Assume that some vertex u of r has valency greater than 2 and let e1 = {u, v1}, e2 = {u, v2}, e3 = {u, v3} be distinct edges. Then x = (e1, e2, e3) is a 2-arc in L(r) and there is no 3-arc y of r such that Ls(y) = x. In general, for s = 3a + b > 4 with a > 1 and b e {0,1,2}, we concatenate a copies of x to form an (s — 1)-arc of L(r): namely (x°) if b = 0; (x°, e1) if b = 1; (x°, e1, e2) if b = 2. This (s — 1)-arc does not lie in the image of Ls. Thus each vertex of r has valency at most 2. If all vertices have valency 2 then r = Cm for some m > 3, since r is connected. So suppose that some vertex u of r has valency 1. Since r is connected and each other vertex has valency at most 2, it follows that r = Pn for some n > s. (2) Let a = (v0,..., vs) be an s-geodesic of r and let Ls(a) = (e0,..., es-1) as above. If s = 2, then Ls(a) is a 1-arc, and hence a 1-geodesic of L(r). Suppose that s > 3 and Ls(a) is not an (s - 1)-geodesic. Then dL(r)(e0, es-1) = r < s - 1 and there exists an r-geodesic r = (f0, f1,..., fr_1, fr) with f0 = e0 and fr = es-1. Since s > 3 and a is an s-geodesic, it follows that {v0, v1} n {vs-1, vs} = 0, that is, e0 and es-1 are not adjacent in L(r). Thus r > 2. Since r is an r-geodesic, it follows that the consecutive edges fi_1, fi, fi+1 do not share a common vertex for any 1 < i < r - 1, otherwise ( f0,..., fi_1, fi+1,..., fr ) would be a shorter path than r, which is impossible. Hence we have fh = {uh, uh+1} for 0 < h < r. Then (u1, u2,..., ur) is an (r - 1)-path in r, {u1} = e0 n f1 C {v0,v1} and {ur} = fr-1 n es-1 C {vs-1, vs}. It follows that dr(v0, vs) < dr(u1,ur ) + 2 < r +1 < s, contradicting the fact that a is an s-geodesic. Therefore, Ls(a) is an (s - 1)-geodesic of L(r). (3) Let 2 < s < diam(L(r)) + 1 and Gs-1 be the set of all (s - 1)-geodesics of L(r). If s = 2, then by part (1), each 1-geodesic of L(r) lies in the image Im(L2), and hence G1 C Im(L2). Now suppose inductively that 2 < s < diam(L(r)) and Gs-1 C Im(Ls). Let e = (e0, e1, e2,..., es) be an s-geodesic of L(r). Then e' = (e0, e1, e2,..., es-1) is an (s - 1)-geodesic of L(r). Thus there exists an s-arc a of r such that Ls(a) = e', say a = (v0, v1,..., vs). Since es is adjacent to es-1 = {vs-1, vs} but not to es-2 = {vs-2, vs-1} in L(r), it follows that es = {vs,x} where x {vs-2,vs-1}. Hence b = (v0, v1,..., vs, x) is an (s + 1)-arc of r. Further, Ls+1(b) = e. Thus Im(Ls+1) contains all s-geodesics of L(r), that is, Gs C Im(Ls+1). Hence the first part of (3) is proved by induction. Now we prove the second part. Suppose first that for every s-arc a of r, Ls (a) is an (s - 1)-geodesic of L(r). Let g := girth(r). If s = 2, as g > 3, then g > 2s - 2. Now let s > 3. Assume that g < 2s - 3. Then r has a g-cycle b = (u0, u1, u2,..., ug-1, ug) with ug = u0. It follows that Lg(b) forms a g-cycle of L(r). Thus the sequence b' = (u0, u1,..., us) (where we take subscripts modulo g if necessary) is an s-arc of r and Ls(b') = (e0, e1,..., es-1) involves only the vertices of Ls(b). This implies that dL(r)(e0, es-1) < 2 < ^^ < s - 1, that is, Ls(b') is not an (s - 1)-geodesic, which is a contradiction. Thus, g > 2s - 2. Conversely, suppose that g > 2s — 2. Let a :— (vo, v1, v2,..., vs) be an s-arc of r. Then Ls(a) = (e0, e1, e2,..., es-1) is an (s - 1)-arc of L(r) by part (1). Let a' := (v0, v1, v2,..., vs-1). Since g > 2s - 2, it follows that a' is an (s - 1)-geodesic, and hence by (2), Ls-1(a') = (e0, e1, e2,..., es-2) is an (s - 2)-geodesic of L(r). Thus z = dL(r)(e0, es-1) satisfies s - 3 < z < s -1. There is a z-geodesic from e0 to es-1, say f = (e0, f1, f2,..., fz-1, es-1). Further, by the first part of (3), there is a (z + 1)-arc b = (u0, u1,..., uz, uz+1) of r such that £z+1(b) = f and we have e0 = {u0, u1} = {v0, v1} and es-1 = {uz,uz+1} = {vs-1,vs}. There are 4 cases, in columns 2 and 3 of Table 1: in each case there is a given nondegenerate closed walk x of length l(x) as in Table 1. Thus l(x) > g > 2s - 2 and in each case l(x) < s + z - 1. It follows that z > s - 1, and hence z = s - 1. Thus Ls(a) = (e0, e1, e2,..., es-1) is an (s - 1)-geodesic of L(r). (4) This property follows from the definition of Ls. □ Table 1: Four cases of x Case (uo, «1) («z, Uz+1) x l(x) 1 (vo, V1 ) (Vs-1, Vs) (Vs-1, Vs-2, . . . , V2, V1, U2, . . . , Uz-1, Vs-1) s + z - 3 2 (vo, V1 ) (Vs, Vs-1) (Vs, Vs-1, . . . ,V2,V1,U2, . . . , Uz-1, Vs ) s + z - 2 3 (V1, vo) (Vs-1, Vs) (Vs-1, Vs-2, . . . , V2, V1,U1,U2, . . . , Uz-1, Vs-1) s + z - 2 4 (V1, vo) (Vs, Vs-1) (Vs,Vs-1,...,V2,V1,U1,U2,..., Uz-1, Vs ) s + z - 1 Remark 2.5. (i) The map Ls is usually not surjective on the set of (s - 1)-arcs of L(r). In the proof of Theorem 2.4 (1), we constructed an (s - 1)-arc of L(r) not in Im(Ls) for any r with at least one vertex of valency at least 3. (ii) Theorem 2.4 (1) and (3) imply that, for each (s - 1)-geodesic e of L(r), there is a unique s-arc a of r such that Ls(a) = e. The s-arc a is not always an s-geodesic. For example, if r has girth 3 and (v0, v1, v2, v0) is a 3-cycle, then a = (v0, v1,v2) is not a 2-geodesic but L2(a) is the 1-geodesic (e0, e1) where e0 = {v0, v1} and e1 = {v1, v2}. 2.2 Proofs of Theorem 1.1 and Corollary 1.2 Proof of Theorem 1.1. Let r be a connected, regular, non-complete graph of girth g and valency at least 3. Then in particular |V(r)| > 5, and by Lemma 2.1, Aut(r) = Aut(L(r)). Let G < Aut(r) and let 2 < s < diam(L(r)) + 1. Suppose first that G is transitive on the set of s-arcs of r. Then by [3, Proposition 17.2], s < g/2 + 1. Since s - 1 < diam(L(r)), it follows that L(r) has (s - 1)-geodesics and by Theorem 2.4 (3), Im(Ls) is the set of (s - 1)-geodesics of L(r). On the other hand, by Theorem 2.4 (4), G acts transitively on Im(Ls), and hence G is transitive on the set of (s - 1)-geodesics of L(r). Conversely, suppose that s < g/2 + 1 and G is transitive on the (s - 1)-geodesics of L(r). Then by the last assertion of Theorem 2.4 (3), Im(Ls) is the set of (s - 1)-geodesics, and since Ls is injective, it follows from Theorem 2.4 (1) and (4) that G is transitive on the set of s-arcs of r. □ Proof of Corollary 1.2. Let r, g, s be as in Theorem 1.1. Assume that Aut(r) is transitive on the (s - 1)-geodesics of L(r). If s > 7, then by [21], Aut(r) is not transitive on the s-arcs of r and so by Theorem 1.1, s > g + 1. □ 3 2-geodesic transitive graphs that are locally cyclic or locally 2K2 In this section, we prove Theorem 1.3. The proof uses the notion of a clique graph. A maximum clique of a graph r is a clique (complete subgraph) which is not contained in a larger clique. The clique graph C(r) of r is the graph with vertices the maximum cliques of r, and two maximum cliques are adjacent if and only if they have at least one common vertex in r. Proof of Theorem 1.3. Let r be a connected non-complete graph of girth 3 and valency 4, and let A = Aut(T) and v e V(r). Suppose first that r is 2-geodesic transitive. Then r is arc transitive, and so Av is transitive on r(v). Since r is non-complete of girth 3, [T(v)] is neither complete nor edgeless, and so, as discussed before the statement of Theorem 1.3, [r(v)] = C4 or 2K2. If [r(v)] = C4, then it is easy to see that r = O (or see [4, p.5] or [5]). So we may assume that [T(v)] = 2K2. It follows from [8, Theorem 1.4] that r is isomorphic to the clique graph C (S) of a connected graph S such that, for each u e V (S), the induced subgraph [S(u)] = 3Ki, that is to say, S is a cubic graph of girth at least 4 and C (S) is in this case the line graph L(S). Moreover, [8, Theorem 1.4] gives that S = C(r). A cubic graph with girth at least 4 has |V(S)| > 5, so by Lemma 2.1, A = Aut(S). Now we apply Theorem 1.1 to the graph S of girth g > 4. Since r = L(S) is 2-geodesic transitive and 3 < g/2 + 1, it follows from Theorem 1.1 that S is 3-arc transitive. Therefore, r is the line graph of a 3-arc transitive cubic graph. Conversely, if r = O, then it is 2-geodesic transitive, and hence is geodesic transitive as diam(O) = 2. If r = L(S) where S is a 3-arc transitive cubic graph, then by Theorem 1.1 applied to S with s = 3, L(S) is 2-geodesic transitive. This proves the first assertion of Theorem 1.3. To prove the second assertion, suppose first that r is geodesic transitive. Then r is distance transitive, and so by Theorems 7.5.2 and 7.5.3 (i) of [4], r is one of the following graphs: O = L(K4), H(2, 3) = L(K3,3), or the line graph of the Petersen graph, the Heawood graph or Tutte's 8-cage. We complete the proof by showing that all these graphs are geodesic transitive. As noted above, O is geodesic transitive; by [7, Proposition 3.2], H(2,3) is geodesic transitive. It remains to consider the last three graphs. Let S be the Petersen graph and r = L(S). Then S is 3-arc transitive, and it follows from Theorem 1.1 that r is 2-geodesic transitive. By [4, Theorem 7.5.3 (i)], diam(r) = 3 and |r(w) n r3(u)| = 1 for each 2-geodesic (u, v, w) of r. Thus r is 3-geodesic transitive, and hence is geodesic transitive. Let S1 be the Heawood graph and S2 be Tutte's 8-cage. Then S1 is 4-arc transitive and S2 is 5-arc transitive, and hence by Theorem 1.1, L(S1) is 3-geodesic transitive and L(S2) is 4-geodesic transitive. By [4, Theorem7.5.3 (i)], diam(L(S1)) = 3 anddiam(L(S2)) = 4, and hence both L(S1) and L(S2) are geodesic transitive. □ Finally, we prove Corollary 1.4. Proof of Corollary 1.4. Let r be a connected non-complete locally cyclic graph. If r is 2-geodesic transitive, then it is regular of valency n say. As discussed in the introduction, n = 4 or 5. If n = 4, then we proved in Theorem 1.3, that r is isomorphic to the octahedron and that the octahedron is indeed 2-geodesic transitive. If n = 5, then by [7, Theorem 1.2], r is isomorphic to the icosahedron, and this graph is 2-geodesic transitive. □ Acknowledgements This article forms part of the second author's Ph.D thesis at UWA under the supervision of other authors. The authors are grateful to Sandi Malnic for suggesting that we consider s-geodesic transitivity for locally cyclic graphs. They also thank the anonymous referees for valuable suggestions on the exposition. References [1] L. Babai, Automorphism Groups, Isomorphism, Reconstruction, Handbook of Combinatorics, the MIT Press, Cambridge, Massachusetts, Amsterdam-Lausanne-New York, Vol 2, (1995), 1447-1540. [2] R. W. Baddeley, Two-arc transitive graphs and twisted wreath products. J. Algebraic Combin. 2 (1993), 215-237. [3] N. L. Biggs, Algebraic Graph Theory, Second ed., Cambridge University Press, Cambridge, (1993). [4] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer Verlag, Berlin, Heidelberg, New York, (1989). [5] A. M. Cohen, Local recognition of graphs, buildings, and related geometries, in: W. M. Kantor, R. A. Liebler, S. E. Payne, E. E. Shult (eds.), Finite Geometries, Buildings, and related Topics, Oxford Sci. Publ., New York. 19 (1990), 85-94. [6] A. Daneshkhah, A. Devillers and C. E. Praeger, Symmetry properties of subdivision graphs, Discrete Math. 312 (2012), 86-93. [7] A. Devillers, W. Jin, C. H. Li and C. E. Praeger, On distance, geodesic and arc transitivity of graphs, preprint, 2011, available at http://arxiv.org/abs/1110.2235. [8] A. Devillers, W. Jin, C. H. Li and C. E. Praeger, Local 2-geodesic transitivity and clique graphs, in preparation. [9] A. A. Ivanov and C. E. Praeger, On finite affine 2-arc transitive graphs, European J. Combin. 14 (1993), 421-444. [10] M. Juvan, A. Malnic and B. Mohar, Systems of curves on surfaces, J. Combin. Theory Ser. B 68 (1996), 7-22. [11] H. D. Macpherson, Infinite distance transitive graphs of finite valency, Combinatorica 2 (1982), 63-69. [12] A. Malnic and B. Mohar, Generating locally cyclic triangulations of surfaces, J. Combin. Theory Ser. B 56 (1992), 147-164. [13] A. Malnic and R. Nedela, K-Minimal triangulations of surfaces, Acta Math. Univ. Comenianae LXIV 1 (1995), 57-76. [14] M. J. Morton, Classification of 4 and 5-arc transitive cubic graphs of small girth, J. Austral. Math. Soc. A 50 (1991), 138-149. [15] C. E. Praeger, An O'Nan Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. 47 (1993), 227-239. [16] C. E. Praeger, On a reduction theorem for finite, bipartite, 2-arc transitive graphs, Australas. J. Combin. 7 (1993) 21-36. [17] C. Thomassen and W. Woess, Vertex-transitive graphs and accessibility, J. Combin. Theory Ser. B 58 (1993), 248-268. [18] W. T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (1947), 459-474. [19] W. T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959), 621-624. [20] R. Weiss, s-transitive graphs, Algebraic methods in graph theory, Vol. I, II, (Szeged, 1978), Colloq. Math. Soc. Janos Bolyai, 25, North-Holland, Amsterdam-New York, (1981), 827-847. [21] R. Weiss, The non-existence of 8-transitive graphs, Combinatorica 1 (1981), 309-311. [22] A. Y. Wu and A. Rosenfeld, Geodesic visibility in graphs, J. Information Sciences 108 (1998), 5-12. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 21-23 Johnson graphs are Hamilton-connected Brian Alspach School of Mathematical and Physical Sciences University of Newcastle Callaghan, NSW 2308, Australia Received 15 December 2011, accepted 15 January 2012, published online 1 June 2012 Abstract We prove that the Johnson graphs are Hamilton-connected and apply this to obtain that another family of graphs is Hamilton-connected. Keywords: Hamilton path, Johnson graph, Hamilton-connected. Math. Subj. Class.: 05C70 1 Main Result The Johnson graph J(n, k), 0 < k < n, is defined by letting the vertices correspond to the k-subsets of an n-set, where two vertices are adjacent if and only if the corresponding k-subsets have exactly k - 1 elements in common. A graph is Hamilton-connected if for any pair of distinct vertices u, v there is a Hamilton path whose terminal vertices are u and v. The graph with a single vertex is trivially Hamilton-connected. In a recent paper [1], I needed a certain graph to be Hamilton-connected. This graph, defined below, contains vertex-disjoint Johnson graphs. The result I needed is embodied in the corollary below. Theorem 1.1. The Johnson graph J (n, k) is Hamilton-connected for all n > 1. Proof. For ease of exposition, instead of talking about the vertex corresponding to a subset, we shall simply treat the subsets as if they are vertices so that we use equality notation between vertices and sets. The graphs J (n, k) and J (n, n - k) are isomorphic via the mapping that takes a k-subset to its complement. The graphs J(n, 0) and J(n,n), n > 1, are isomorphic to the single vertex K1 and trivially Hamilton-connected. The graphs J (n, 1) and J (n, n — 1), n > 1, are isomorphic to the complete graph Kn. Complete graphs certainly are Hamilton-connected. We proceed by double induction and when considering J(n, k), the induction hypotheses are: J(m, k') is Hamilton-connected whenever k' < k and m > k', or J(m, k) is E-mail address: brian.alspach@newcastle.edu.au (Brian Alspach) Hamilton-connected whenever m < n and m > k. As noted above, J(n, 1) is Hamilton-connected for all n > 1. For a fixed k we start with J(k, k) and then proceed by going from J(n - 1, k) to J(n, k). Thus, the induction hypotheses make sense. If k < n < 2k — 1, then n — k < k so that J (n, n — k) is Hamilton-connected by hypothesis. This, in turn, implies that J(n, k) is Hamilton-connected because J(n, k) and J (n, n — k) are isomorphic. Thus, it follows that J (n, k) is Hamilton-connected for all n satisfying k < n < 2k — 1. For the remaining cases we need to actually show how to find appropriate Hamilton paths. The symmetric group Sn acts in the obvious way on the vertex set of J(n, k). This action is transitive so that it suffices to find a Hamilton path from the vertex u = {1, 2,3,..., k} to any other vertex. Let v = {ai, a2,..., ak} be an arbitrary vertex. If there is an element x of {1, 2,..., n} belonging to neither of the sets, we may relabel elements so that n is missing from both sets. Thus, both k-sets are subsets of {1, 2,..., n — 1}. By induction there is a Hamilton path from u to v in J(n — 1, k). Because the vertices that are adjacent along that path also are adjacent in J(n, k), let P' be the corresponding path from u to v in J(n, k). The path P' contains all the vertices corresponding to k-subsets that do not contain n. Let wi = {yi,y2,... ,yk-i,Vk} and w2 = {yi, V2,..., Vk-i, zk} be two adjacent vertices on P'. The vertex wi is adjacent to the vertex w3 = {y2,..., yk-i, yk, n}, and the vertex w2 is adjacent to the vertex w4 = {y2,..., yk-i, zk, n}. The subgraph X induced by J(n, k) on all the subsets containing n clearly is isomorphic to J(n — 1, k — 1). Thus, there is a path from w3 to w4 spanning all the vertices of X. Thus, remove the edge of P' between wi and w2, add the edges wiw3 and w2w4, and then add the path from w3 to w4 spanning X. The resulting path is a Hamilton path in J(n, k) with u and v as terminal vertices. If n > 2k, then there always is an element x missing both subsets and the preceding argument establishes that J(n, k) is Hamilton-connected. If n = 2k, there is exactly one subset that fails the criterion, namely, the complement of {1, 2,..., k}. So we need to find a Hamilton path in J(2k, k) from u to its complement. Consider the k-subsets of {1, 2,..., 2k} not containing the element 2k. The subgraph induced by J(2k, k) on this collection of subsets is isomorphic to the graph J(2k — 1, k). It is Hamilton-connected by induction so that there is a Hamilton path from u to w = {1, 2,..., k — 1, 2k — 1}. Let P be the copy of this path in J(2k, k). Now consider all the k-subsets of {1,2,..., 2k} that contain the element 2k. The subgraph Y' induced on this collection of sets is isomorphic to J(2k — 1, k — 1) so that it has a spanning path from {1,2,..., k — 1, 2k} to {k + 1, k + 2,..., 2k}. Because the intermediate terminal vertices on the two paths are adjacent, we have a Hamilton path in J(2k, k) from u to its complement. This completes the proof. □ The corollary below is the real target of this short paper. We need to define a particular graph first. Let A = {ai, a2,..., am} be a non-empty subset of {0,1, 2,..., n} such that the elements are listed in the order ai < a2 < • • • < am. We define the graph QJ(n, A) in the following way. For each až G A, we include a copy of the Johnson graph J (n, až). Thus far the Johnson graphs are vertex-disjoint. We then insert edges between J (n, až) and J(n, ai+i), for each i, using set inclusion, that is, we join an aj-subset Si and an ai+i-subset S2 if Si is contained in S2. The graph QJ(n, A) can be pictured as having levels made up of Johnson graphs with edges between successive levels based on set inclusion. Corollary 1.2. The graph QJ (n, A) is Hamilton-connected for all n > 1. Proof. If A is a singleton set, then QJ(n, A) is a Johnson graph and the result follows from Theorem 1.1. Hence, we assume that A has at least two elements. Suppose that u and v are two vertices of QJ(n, A) lying at different levels, where u has cardinality a^ v has cardinality a j, and ai < a j. Construct a path starting at u that spans the vertices at level ai and terminates at an arbitrary vertex ui at level a^ Choose a neighbor ui+1 of uj at level ai+1 making certain it is distinct from v if j = i + 1. Then add the edge from u j to ui+1 followed by a path spanning the vertices at level ai+1. If v happens to lie at this level make certain the path terminates at v. Otherwise, the path can terminate at any vertex at level ai+1. It is obvious how to continue this until we have a path starting at u, terminating at v, and spanning all the vertices at levels ai, ai+1 up through level a j. If this happens to be all the levels of QJ (n, A), then we have found a Hamilton path joining u and v. If we are missing levels, we then continue as follows. If there are missing levels above level a j, then remove an edge xy of the current path at level aj and take distinct neighbors xX and y' of x and y, respectively, at level aj+1. Then extend to a larger path by taking a path joining x' and y' spanning all the vertices at level aj+1. If x and y don't have distinct neighbors at level aj+1, then aj+1 = n and the level is the singleton vertex w = {1, 2,3,..., n} which is adjacent to everything at level aj so that we replace the edge xy of the path with the 2-path xwy. It is obvious how to continue adding the vertices one level at a time until we finish with the top level. We also can do the analogous extension with the levels below ai until we achieve a Hamilton path in Q(n, A) that has u and v as terminal vertices. If u and v are at the same level. Then we start with a path spanning level ai that has u and v as terminal vertices. We then extend the path through the other levels as above. This completes the proof. □ Corollary 1.2 allows us to process a variety of collections of sets of different cardinalities where we can move from one set to another either by a revolving door operation or restricted inclusions. This is what was required in [1]. References [1] B. Alspach, Hamilton paths in Cayley graphs on Coxeter groups: I, preprint. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 25-35 Nonorientable regular maps over linear fractional groups Gareth A. Jones University of Southampton, Southampton, U.K. Martin MaCaj Comenius University, Bratislava, Slovakia Jozef Siran Open University, Milton Keynes, U.K. Slovak University of Technology Bratislava, Slovakia Received 28 October 2011, accepted 22 January 2012, published online 1 June 2012 Abstract It is well known that for any given hyperbolic pair (k, m) there exist infinitely many regular maps of valence k and face length m on an orientable surface, with automorphism group isomorphic to a linear fractional group. A nonorientable analogue of this result was known to be true for all pairs (k, m) as above with at least one even entry. In this paper we establish the existence of such regular maps on nonorientable surfaces for all hyperbolic pairs. Keywords: Regular map, linear fractional group. Math. Subj. Class.: 05C10, 05C25, 20G99 1 Introduction A map on a compact, orientable surface is orientably regular if the group of all orientation preserving automorphisms of the map is transitive, and hence regular, on darts of the map. A map on a compact, nonorientable surface is regular if its automorphism group is transitive, and hence regular, on flags of the map. In either case, such maps have all vertices of the same degree and all faces of the same length; if these quantities are k and m we speak of a map of type {m,k}. The type is said to be hyperbolic if 1/k + 1/m < 1/2. Regarding type, the following basic fact was rediscovered a number of times in the past. E-mail addresses: g.a.jones@soton.ac.uk (Gareth A. Jones), macaj@fmph.uniba.sk (Martin Macaj), j.siran@open.ac.uk (Jozef Siran) Theorem 1.1. For every hyperbolic pair (k, m) there exist infinitely many orientably regular maps of type {m, k}. A brief history of the development around this result together with a list of various proofs can be found in [9]. A particularly important way of proving Theorem 1.1 follows from [8] and implies that all such maps can be chosen to have the orientation preserving automorphism group isomorphic to a linear fractional group over a finite field. It is quite surprising that a nonorientable analogue of Theorem 1.1 has not been considered. A proof might follow from the study of generation of symmetric and alternating groups by pairs of permutations of given order in [6], but this work does not appear to have a corresponding follow-up and it is not our intention to do so. Instead, motivated by the result of [8] mentioned above, we will be interested in possibilities to prove a stronger form of a nonorientable analogue of Theorem 1.1 for maps with automorphism group isomorphic to a linear fractional group over a finite field. In fact, this has already been done for three quarters of the cases by the third author in [11] where it is shown that for any hyperbolic pair (k, m) with at least one even entry there are infinitely many nonorientable regular maps of type {m, k} with automorphism group isomorphic to PGL(2, F) for suitable finite fields F, but the case when both k and m are odd was left untouched. In this note we fill in this gap, establishing thus the existence of an infinite number of nonorientable regular maps of type {m, k} with automorphism group isomorphic to a linear fractional group for any given hyperbolic type {m, k}. The main results are presented in Section 4, preceded by background information summed up in Section 2 and auxiliary number theoretic results in Section 3. 2 Preliminaries Foundations of the theory of regular maps have been laid in [4] and [2] and in what follows we just briefly review a few basic facts; for surveys we recommend [7] and [10]. Orientably regular maps of type {m, k} can be identified with their orientation preserving automorphism groups and these are in a one-to-one correspondence with the finite groups G presented in the form G = (r, s; rk = sm = (rs)2 = ... = 1) (2.1) where r and s represent a k-fold rotation about a fixed vertex of the map and an m-fold rotation about the centre of a face incident with the vertex. In particular, we require that k, m and 2 are the true orders of r, s, and rs, respectively. Vertices, faces and edges of the orientably regular map Mor (G) corresponding to a presentation of a group G as in (2.1) can be identified with left cosets of the cyclic subgroups (r), (s) and (rs), with incidence determined by non-empty intersection; the group G then acts as the orientation preserving automorphism group of Mor(G) by left multiplication. Regular maps on nonorientable surfaces are also in a one-to-one correspondence with presentations of finite groups as in (2.1) but satisfying the extra condition that G contains an involution t such that trt = r-1 and tst = s-1. This time, the nonorientable regular map Mnor(G) corresponding to such a group G has vertices, faces and edges identified with the left cosets of the dihedral subgroups (r,t), (s,t) and (rt,ts). Incidence is again determined by non-empty intersection and G acts as the automorphism group of Mnor(G) by left multiplication. Thus, if a group G with presentation (2.1) admits an inner automorphism induced by an involution and inverting r and s, the above correspondences allow one to associate two maps with G, namely, Mor(G) and Mnor(G). To remove this ambiguity in what follows, for a group G as in (2.1) we define the map M (G) by letting M (G) = Mnor(G) if G has an inner automorphism induced by an involution, inverting both r and s, and M (G) = Mor(G) if G has no such inner automorphism. As stated in the Introduction we will be interested in regular maps of a given type with automorphism group isomorphic to a linear fractional group. We begin by recalling the characterisation of such automorphism groups from [8]; for a much more detailed proof we refer to [3]. Proposition 2.1. Let (k, m) be a hyperbolic pair and let K be an algebraically closed field of a prime characteristic p coprime to km. Let e and n be primitive (Sk)th and (Sm)th roots of unity in K, where S =1 if p = 2 and S = 2 if p > 2. Let D = -(£2 + + n2 + n-2) and let R = ± e o o e-1 and s = ±(e — e-1) 1 -1 -(n+n-1)e-1 d i (n + n-1)e be elements of PSL(2, K). Then, (a) the orders of R, S, and RS in PSL(2, K) are k, m, and 2, respectively, and (b) if G is a subgroup of PSL(2, K) with presentation (2.1), then there exist primitive (Sk)th and (Sm)th roots of unity such that G is conjugate to the subgroup generated by the matrices R and S as above. It is therefore sufficient to study the groups G(£, n) = R S} with R and S as above. Necessary and sufficient conditions for G(£, n) to give rise to a nonorientable regular map were given in [3]. Here we present an excerpt sufficient for our purposes. Theorem 2.2. Let (k, m) be a hyperbolic pair and let K be an algebraically closed field of a prime characteristic p relatively prime to both k and m. Let e and n be primitive (Sk)th and (Sm)th roots of unity in K, where S = 1 if p = 2 and S =2 if p > 2. Let e = e(k, m) be the smallest positive integer j such that n | (pj — en)/S for each n e {k, m} and some £n e {+1, —1}. Then: (1) if e is even, e = 2f, then G(e,n) is isomorphic to PGL(2,pf ) if and only if the quantity D = —(e2 + e-2 + n2 + n-2) is not equal to zero, and either (a) there is an even entry n e {k, m} and an e e {+1, —1} such that n divides (pf — e) but 2n does not, while the other entry divides (pf — e')/2 for some e' e {+1, —1}, or (b) both k and m are even and for any n e {k, m} there exists an e'n such that n is a divisor of (pf — e'n) but 2n is not; (2) G(e, n) is isomorphic to PSL(2, pe) if and only if D = 0 and either e is odd, or the pair (k, m) together with an even e do not satisfy any of the above conditions (a) and (b); and (3) if D = 0, the map M(G(e,n)) is nonorientable if and only if either e = 2f and G(e,n) = PGL(2,pf ), or if G(e,n) = PSL(2,pe) and D is a square in GF(pe); in particular, in the last case M (G (e, n)) is always nonorientable if p = 2 and both k and m are odd. From the last part of this result one sees that to obtain nonorientable regular maps it is, for example, sufficient to make sure that e — 2f, D — 0 and G(£, n) — PGL(2,pf ). In [11] it is shown that this can be guaranteed whenever at least one of k and m is even: Theorem 2.3. Let (k, m) be a hyperbolic pair with at least one even entry. Then, there is an infinite number of finite, nonorientable, regular maps of type {m, k} with automorphism group isomorphic to PGL(2, F) for suitable finite fields F. To be able to extend this result to the case when both k and m are odd, by Theorem 2.2 one can only hope to establish the existence of infinitely many regular maps of type {m, k} with automorphism group isomorphic to PSL(2, F) for suitable finite fields F. In particular, by part (3) of Theorem 2.2, to achieve this we need to make sure that for an infinite number of primes p one can select primitive 2k-th and 2m-th roots of unity £ and n in GF(pe) in such a way that the quantity D — — (£2 + £-2 + n2 + n-2) is a square in GF(pe), where e — e(k, m); note that e depends on p as well but this dependence is not shown in our notation. Observe that if k and m are odd, £2 and n2 are primitive k-th and m-th roots of unity, respectively. Thus, in this case D has the form D — — (wk + wm) where w„ denotes the sum of an n-th primitive root of unity and its reciprocal in GF (pe). In what follows we will investigate such quantities in general, first over the field of complex numbers and subsequently over finite fields by considering factor fields of rings of algebraic integers. 3 Auxiliary results involving complex roots of unity Let Cn denote any primitive n-th root of unity, but this time taken in the field C of complex numbers unless stated otherwise. It is known that all the primitive n-th roots of unity are conjugate over the rationals Q and their common minimal polynomial is the n-th cyclo-tomic polynomial $n of degree y(n), the value of the Euler totient function at n. By wn we denote any number of the form Zn + Z-1 ; these quantities are again conjugate over Q and their common minimal polynomial will be denoted by (x). It is well known that if n > 2, then Xv(n)/2^n(x + X-1)— $n(x). (3.1) Finally, let Un denote the set of all primitive n-th roots of unity in C and let Un stand for the set of all the corresponding quantities wn. We continue with some observations. From the fact that $1(x) — x — 1, $p(x) — 1+ x +-----+ xp— 1 and $pn(x) — $n(xp) ifp|n and $pn(x) — $n(xp)/$n(x) otherwise, we obtain the following auxiliary result by easy calculations. Lemma 3.1. Let $n(x) be the n-th cyclotomicpolynomial. Then, $1(1) — 0, (1) — p for p prime and k > 0, and $n(1) — 1 otherwise. Also, $1( —1) — —2, $2( —1) — 0, $2pfc ( —1) — p for p prime and k > 0, and $n( —1) — 1 otherwise. □ With the help of these facts we obtain our basic result on products of the quantities — (wfc + wm) for any G U k and G U m. Proposition 3.2. Let k, m be odd positive integers and let P (k, m)— JJ JJ — (wfc + wm) . wfc eukeum Then, P (1,1) — —4, P (k, k)2 — (—2)^(k) for k > 3, and P (k, m)2 — 1 otherwise. Proof. Obviously P (k, m) = P (m, k) and we will therefore assume that k > m in what follows. The values of P (k, m) for k, m < 2 are trivial. If k > 3 and m =1, then P (k, 1) = n Ukeuk -(2 + "k ) = (-2) = (-1 )-^)/2$fc (-1) = (-1)^V2 and so P (k, 1)2 = 1. For the remaining part of the proof we assume that k > m > 1. By the properties of the polynomials ^m(x) = (x - Wm) UmEU m we obtain, for any wk = Ck + C-1 € U k, the equality H -K + Wm) = ^m(-Wfc ) = (-Cfc )-V(m)/2$m(-Cfc ) . UmEU m Let U'k be a subset of Uk of cardinality y(k)/2 such that Uk = {Ck + Z- ; Cfc € U'}. The previous computation then implies that P (k, m) = n (-Cu )-v(m)/2*m(-Cfc ) . Cfc Euk Extending the product above from Uk' to Uk means squaring the last equation; combining this with the fact that the product of all the (even number of) k-th primitive roots of unity is equal to 1 we obtain P (k, m)2 = n (-Zk )-V(m)/2 ^m(-Cfc )= n ^m(-Cfc). Cfc E U k Cfc E U k Invoking the well known identity $m(x) = f]d|m(xd - 1)^(m/d), where ^ is the Moebius function, we have *m(-Zfc ) = n(-Zd - 1)M(m/d) . d| m This product is non-zero since both k and m, and hence all the divisors d, are odd and so (-Cfc )d = 1; note also that the divisors satisfy d < k because of the assumption m < k. Let us analyze the system of powers U = (Cd; Cfc € Uk ) appearing in the last equality. For any positive divisor d of m let n(d) = k/(d, k) and r(d) = y>(k)/<^(n(d)); of course, both quantities depend on k as well. It can now be seen that the system U is a collection, for any d dividing m, of primitive n(d)-th roots of unity, each repeated r(d) times. With the help of all these facts together with $t (x) = f]^Ejjt (x - Ct) evaluated at x = -1 we successively obtain p (k, m)2 = n n(-cd - 1)M(m/d)=n n(-1 - cd )m( Zk EU k d|m d|m Zk E Uk =n n (-1 - Cn(d))r(d)M(m/d)=n($n(d)(-1))r(d)M(m/d). d|m Zn(d)EUn(d) d|m As all the values of n(d) are odd here, we have $n(d)(-1) = 1 if d < k and $n(d)(-1) = -2 if d = k, where the second possibility occurs if and only if m = k and then r(d) = <^(k) and ^(m/d) = 1. We conclude that for k > 3 we have P (k, k)2 = (-2)v(k), and P (k, m)2 = 1 if 1 < m < k. This completes the proof. □ A different and more powerful approach to the investigation of the quantity D from Theorem 2.2 relies on some known facts on algebraic integers in algebraic number fields. We refer to [1] as a suitable introductory reference and recall here just a few basic concepts and results. Let K be an algebraic number field, that is, an extension of Q of a finite degree. Let O = OK be the ring of algebraic integers in K. The ring O is known to be a Dedekind domain, but apart from a few facts the theory of such domains will not be needed. A basic property of O is that every non-zero ideal J C O has a finite index [O : J]. Without going into too much detail we recall that the the index [O : J] is the norm N (J ) of J. Another important property of O is that any prime ideal J C O is maximal. Thus, for any such J the quotient ring O/J is a finite field and so there exists a unique rational prime p such that N(J) = pj for some j e {1,2,..., d}, where d = [K : Q] is the degree of the extension. Further, it is known that K admits exactly d distinct injective homomorphisms a^,...,ad into C. The norm N (z) of any element z e K is defined as the product N(z) = 3. Since algebraic integers have integral norm, it follows that A is a unit in O if k = m. In the case when k = m and a = ß, the absolute value of the norm of —2(a2 + a-2) is equal to (—2)^(fc)/2, and therefore for a = ß the absolute value of the norm of A must be 1. □ Returning to our main theme, until the end of this section we will assume that (k, m) is a fixed hyperbolic pair with no restriction on the parity of the two entries. We begin by an elementary observation that will turn out to be crucial later. Lemma 3.4. The quantity A — n2 is never a unit in O for any integer n > 2. Proof. We recall the known fact that K is isomorphic to the cyclotomic field Q[y], where Y = cos( ) + i sin() is a primitive i-th root of unity for i = lcm{2, k, m}. The conjugates of 7 over K have the form cos( 2n j ) + i sin( 2n j ), where 1 < j < I and (j,£) = 1. All the ip(£) distinct injective homomorphisms at : K ^ C preserve the rationals pointwise. Since the explicit form of the conjugates of 7 over K implies that MA) | < 4, we have | at (A - n2)| = | at (A) - n21 > n2 - |at(A)| > n2 - 4 for any t such that 1 < t < y(l). Thus, by the definition of the norm, for n > 2 we have |N(A - n2)\ > 1, which means that A - n2 is not invertible in O. □ It is useful to realise that our considerations before Lemma 3.1 did not depend on the parity of k and m and hence we may use them in what follows. Observe that in the general case we want to deal with, the value of A could be equal to zero in K, which happens precisely if iß e {±a, ±a-1}. If, however, {k, m} is a hyperbolic pair, it is easy to see that we can choose a and ß avoiding this condition. Keeping to the notation introduced above, for any n > 3 let I = In be a maximal ideal in O containing the element A - n2 and let p = pn be the characteristic of the field F = O/I. Letting £ = a + I, n = ß + I, and D = A + I, we have: Lemma 3.5. If n is relatively prime to N (A), then the element D = - (£2 +£-2 +n2 +n-2) is a non-zero square in F and p does not divide n. Moreover, if p is not a divisor of 2 km, then £ and n are primitive 2k-th and 2m-th root of unity in F. Proof. Since A - n2 e I, that is, A + I = n2 +1, the element D = A + I is a square in F .As p e I and I is a prime ideal, by our earlier remarks on norms of elements and ideals of the Dedekind ring O the condition A e I is equivalent to each of the conditions n2 e I, n e I, and p|n. Hence p divides both n and N(A), contrary to our assumption on their relative primeness. It is obvious that £ is a 2k-th root of unity in F. Assume that au - 1 e I, where au is a primitive c-th root of unity in C for a proper divisor c of 2k. As the ideal generated by the algebraic integer au - 1 is contained in I, the norm of I divides the norm of au - 1, which implies that the norm of au - 1 is divisible by p. On the other hand, all conjugates of au are c-th primitive roots of unity in C. Arguments analogous to those used in the proof of Proposition 3.2 imply that, up to sign, the norm of au - 1 is a power of $c(1). Thus, by Lemma 3.1, c is a power of p, contrary to the assumption that p \ 2k. It follows that £ is a primitive 2k-th root of unity in F. By the same token, n is a primitive 2m-th root of unity in F. □ By suitably varying the parameter n one obtains an infinite sequence of primes as in Lemma 3.5. Lemma 3.6. If A = 0, then there exists an infinite set of values n and an infinite sequence of prime ideals In of O containing the element A - n2 such that the fields O/In have pairwise distinct prime characteristic pn. Proof. Referring to the way the primes pn have been introduced for any n > 2, let us define an infinite sequence nj of integers by letting ni = 2|N(A) | + 1 and nj = nj=i pn for j > 1. Applying Lemma 3.5 inductively we deduce that pnj does not divide nj for any j > 1. By our construction, for any j > 2 the prime pnj differs from all the previous primes pni for i < j. □ 4 Results Two immediate consequences in the direction of our interest can be obtained by exploring earlier results. Firstly, there is a much more general version of Theorem 2.2 in which the prime p is not necessarily coprirne to k and m, in particular, covering the case when both k and m are equal to p and G = PSL(2,p); see Propositions 3.1, 3.2, 4.6 and 6.1 of [3]. In order to avoid a rather long re-statement of these facts we invite the reader to check that part (2) of Proposition 6.1 combined with Proposition 3.1 of [3] imply: Theorem 4.1. If p is a prime congruent to 1 mod 4, then there exists a nonorientable regular map of type (p,p) with automorphism group isomorphic to PSL(2,p). □ Secondly, if both k and m are odd and p = 2, part (3) of Theorem 2.2 directly yields the following result, where e = e(k, m) is as introduced in the statement of Theorem 2.2. Theorem 4.2. Let (k, m) be a hyperbolic pair consisting of odd entries. Then there is a nonorientable regular map of type {m, k} with automorphism group isomorphic to PSL(2, 2e) for e = e(k, m). □ Together with the earlier findings this gives at least an existence result of the sought kind on regular maps over linear fractional groups. Corollary 4.3. For any hyperbolic pair (k, m) there exists a nonorientable regular map of type {m, k} with automorphism group isomorphic to a linear fractional group over a finite field. □ In the light of Theorem 2.3, the question of existence of an infinite number of such maps of any given type hyperbolic type is settled by the following result. Although we are interested mainly in the case when k and m are odd, we give a more general formulation which yields an alternative proof of Theorem 2 of [11]. Theorem 4.4. For every hyperbolic pair (k, m) there is an infinite number of finite, nonorientable, regular maps of type {m, k} with automorphism group isomorphic to PSL( 2, F ) or PGL( 2, F ) for suitable finite fields F. Proof. We will refer to the notation introduced in Section 3. For a fixed hyperbolic pair (k, m) and a non-zero A = —(a2 + a-2 + ß2 + ß-2) with let p = pn be any prime from Lemma 3.6 relatively prime to 2km, and let G = G(£, n) be the corresponding group. By Theorem 2.2, G is isomorphic to PSL(2, F) or PGL(2, F'). As D is a square, the corresponding regular map M (G) is nonorientable in both cases. □ We also present two more results based on residue techniques which, although applicable only to a very restricted infinite set of types with both entries odd, may be useful in future investigations. Theorem 4.5. Let k and m be prime powers congruent to 3 mod 4. Then, there exist infinitely many nonorientable regular maps of type {m, k} with automorphism group isomorphic to PSL(2, F) for suitable finite fields F. Proof. Let p be a prime congruent to 1 mod 8 and let e = min{n; k|pn ± 1 and m|pn ± 1}. Then, p does not divide 2km and the equality from Proposition 3.2 holds also in GF(pe), with appropriate interpretation of the primitive roots. Since p = 1 mod 8, the four elements ±1 and ±2 are all quadratic residues in GF(p) and hence also in GF(pe). By Proposition 3.2, the element P (k, m) is a quadratic residue in GF (pe) ; note that in the case k = m it would have been sufficient to assume p = 1 mod 4 to obtain the same conclusion. By our assumptions, both <(k) / 2 and <(m)/2 are odd. The product P (k, m) has therefore an odd number of factors and so at least one of them must be a quadratic residue in GF(pe ). That is, there exist wk g Uk and g Um such that the value D = - (wk +wm) is a square in GF(pe). This gives, by Theorem 2.2 and the remark after Theorem 2.3, a nonorientable regular map of type {m, k}. Since there are infinitely many primes p as above, our result follows. □ Theorem 4.6. Let k and m be odd integers forming a hyperbolic pair such that the number <(k)<(m)/4 is even, that is, at least one of k, m is not a prime power congruent to 3 mod 4. If P (k, m) < 0, then there are infinitely many finite, nonorientable, regular maps of type {m, k} with automorphism group isomorphic to PGL( 2, F ) for suitable finite fields F. Proof. Let p = 3 mod 4 be a prime such that e is odd (e.g., any p = ±1 mod km). If k = m, then P (k, m) = —1 not only in C but also in GF (pe). Similarly if k = m, then we have P (k, m) = (—2)^(fc)/2 both in C and in GF (pe). Note that if k = m, then <(k)/2 is even and 2v(k)/2 is a quadratic residue in GF(p). As p = 3 mod 4 and e is odd, the product P (k, m) is a quadratic nonresidue in both GF (p) and GF (pe). On the other hand, P(k, m) has an even number of factors, and therefore at least one of them is a quadratic residue in GF (pe ). □ 5 Remarks By Theorem 4.4, for any given hyperbolic pair (k, m) there exists an infinite number of nonorientable regular maps of type {m, k} with automorphism group isomorphic to linear fractional groups over finite fields. Our approach was based on developing some results obtained in [3] in the course of analysing regular maps over linear fractional groups. The scope of [3] is, however, broader and covers regular hypermaps. For a general theory of hypermaps and their surface representations we recommend [5]. Here we just recall that a finite regular hypermap of type (k, m, l) can be identified with a finite quotient group of the triangle group T (k, m, I) = (r, s, t; rk = sm = (rs)1 = 1). Thus, regular hypermaps are a natural generalisation of regular maps (corresponding to the case when I = 2). Facts collected in [8, 3] imply that for any hyperbolic triple (k, m, l), that is, such that 1/k + 1/m + 1/1 < 1, there exist infinitely many regular hypermaps of type (k, m, l) on orientable surfaces, with automorphism group isomorphic to a linear fractional group over a finite field. By the theory developed in [3] combined with the findings in this paper, to establish a nonorientable analogue of this result requires analysing conditions under which the quantity A' = 4+(a+a-1)(ß+ß-1)(7+7-1)-(a+a-1)2 -(ß+ß-1)2 -(7+7-1)2, where a, ß and 7 are primitive 2k-th, 2m-th, and 21-th roots of unity in C, projects onto a non-zero square in a quotient field of the ring of algebraic integers of Q[a, ß, 7] generated by a suitable prime ideal; note that for 1 = 2 we have 72 = 1 and 7 + 7-1 =0 and then A' reduces to the quantity A introduced earlier. In fact, methods of Section 3 can be adapted in an obvious way to construct, for any hyperbolic triple (k, m, 1) and for suitable triples (a, ß, 7) of primitive roots of unity as above, an infinite number of nonorientable regular hypermaps of type (k, 1, m) over linear fractional groups. A comparison of Theorems 4.4, 4.5 and 4.6 reveals their different nature. Theorem 4.4 is more universal since it applies to all hyperbolic pairs and it is, in essence, constructive, but it yields no information on the corresponding set of primes. On the other hand, Theorems 4.5 and 4.6 apply to a very restricted set of hyperbolic pairs and are, in essence, existential, but the sets of primes for which they guarantee the existence of nonorientable regular maps have positive density in the set of all primes. This leaves the intriguing question of whether it is possible, for any given hyperbolic pair (k, m), to determine all primes p such that there exists a nonorientable regular map of type {m, k} with its automorphism group isomorphic to a linear fractional group over a field of characteristic p. For possible further interest we present a table of values of the product P (k, m) for odd k, m such that 3 < k, m < 41. Observe that for k = m the table shows negative entries only if both k and m are powers of primes congruent to 3 mod 4. If this observation carries through to all odd k and m, Theorem 4.6 would be applicable only in the case when k = m, and the values 5, 13, 25, 29 and 37 show that this Theorem is not void. 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 2 1 -1 -1 -1 1 1 1 -1 1 -1 1 — 1 1 —1 1 1 1 1 1 1 — 22 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 — 23 -1 -1 1 1 1 -1 1 -1 1 — 1 1 —1 1 1 1 1 1 -1 1 -1 23 -1 1 1 1 -1 1 -1 1 — 1 1 —1 1 1 1 1 1 -1 1 -1 -1 25 1 1 1 -1 1 -1 1 — 1 1 —1 1 1 1 1 1 1 1 1 1 1 — 26 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 24 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 28 1 1 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 29 1 -1 1 —1 1 —1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 26 1 1 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 —211 1 —1 1 —1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 — 210 1 1 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 -1 1 29 1 —1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 —214 1 1 1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1 1 -1 1 —1 1 —215 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 210 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 212 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 —218 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 212 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 220 Acknowledgement Research of the second author was supported by the APVV Research Grants 0111-07 and 0223-10, and the VEGA Research Grants 1/0588/09 and 1/0406/09. The third author acknowledges support by the APVV Research Grants 0104-07 and 0223-10, and the VEGA Research Grants 1/0280/10 and 1/0781/11. Both the second and the third authors also acknowledge the APVV support as part of the EUROCORES Programme EUROGIGA (project GREGAS, ESF-EC-0009-10) of the European Science Foundation. References [1] S. Alaca and K. S. Williams, Introductory Algebraic Number Theory, Cambridge University Press, Cambridge, 2004. [2] R. P. Bryant and D. Singerman, Foundations of the theory of maps on surfaces with boundary, Quart. J. Math. Oxford Ser. 141 (1985), 17-41. [3] M. Conder, P. Potocnik and J. Siran, Regular hypermaps over projective linear groups, J. Australian Math. Soc. 85 (2008), 155-175. [4] G. A. Jones and D. Singerman, Theory of maps on orientable surfaces, Proc. London Math. Soc. 37 (1978), 273-307. [5] G. A. Jones and D. Singerman, Belylfunctions, hypermaps, and Galois groups, Bull. London Math. Soc. 28 (1996), 561-590. [6] Q. Mushtaq and H. Servatius, Permutation representations of the symmetry groups of regular hyperbolic tessellations, J. London Math. Soc. 48 (1993), 77-86. [7] R.Nedela, Regular maps - combinatorial objects relating different fields of mathematics, J. Korean Math. Soc. 38 (2001), 1069-1105. [8] Ch.-H. Sah, Groups related to compact Riemann surfaces, Acta Math. 123 (1969), 13-42. [9] J. Siran, Triangle group representations and their applications to graphs and maps, Discrete Math. 229 (2001), 341-358. [10] J. Siran, Regular maps on a given surface: a survey, Topics in Discrete Mathematics, Algorithms Combin. 26, Springer, 2006, 591-609. [11] J. Siran, Non-orientable regular maps of a given type over linear fractional groups, Graphs and Combinatorics 26 (2010), 597-602. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 37-56 Embeddings of cubic Halin graphs: Genus distributions* Jonathan L. Gross Columbia University, Department of Computer Science NY 10027 USA, New York, USA Received 27 June 2011, accepted 8 April 2012, published online 1 June 2012 Abstract We derive an O (n2)-time algorithm for calculating the genus distribution of a given 3-regular Halin graph G; that is, we calculate the sequence of numbers g0(G), gi (G), g2(G), ... on the respective orientable surfaces S0, Si, S2,... . Key topological features are a quadrangular decomposition of plane Halin graphs and a new recombinant-strands reassembly process that fits pieces together three-at-a-vertex. Key algorithmic features are reassembly along a post-order traversal, with just-in-time dynamic assignment of roots for quadrangular pieces encountered along the tour. Keywords: Genus distribution, Halin graph, partitioned genus distribution, gram embedding, outer-planar graph, topological graph theory. Math. Subj. Class.: 05C10 1 Introduction A Halin graph [20] is constructed from a plane tree T with at least four vertices and no 2-valent vertices by drawing a cycle thru the leaves of T in the order they occur in a preorder traversal of T. Any wheel graph Wn (for n > 3) is a Halin graph. Every Halin graph can be obtained by iterative splitting of the hub of a wheel and of some of the resulting vertices. Although some of the graphs obtained by splitting the hub of a wheel are non-planar, every planar graph so obtained is a Halin graph, since splitting a vertex of a tree yields a tree. The outer cycle of a Halin graph is the cycle corresponding to the traversal of the leaves of the inscribed tree. [Since a Halin graph is 3-connected, its planar embedding is unique up to reversal of orientation, as per Whitney's theorem.] In the Halin graph of Figure 1, the outer cycle has length eight. * This paper was presented as a keynote address at the 7th Slovenian International Conference on Graph Theory in June, 2011, at Lake Bled, Slovenia. E-mail address: gross@cs.columbia.edu (Jonathan L. Gross) Figure 1: A Halin graph for a 14-vertex tree with 8 leaves. Genus distributions Def. The genus distribution for graph G is the sequence Ydist(G) : go(G), gi(G), 92(G), ••• where Qì(G) denotes the number of embeddings of G in the orientable surface Si of genus i. In reckoning the number of embeddings of the graph G in the surface S, we regard two embeddings i : G ^ S and i' : G ^ S as the same if there is an extension of the identity automorphism 1G : G ^ G to an orientation-preserving autohomeomorphism h : S ^ S such that i o h = i' or, equivalently, if the two embeddings of G correspond to the same rotation system (see [19]). Calculating the genus distribution of a graph requires determining not only its minimum genus and its maximum genus, but also the number of embeddings of every possible genus. Table 1 gives the genus distributions of some familiar graphs, each of which is small enough that its genus distribution can be calculated by hand using ad hoc methods. Table 1: Genus distributions of some familiar graphs. graph G go(G) Si (G) 92(G) 93(G) 94(G) ••• K4 2 14 0 0 0 ... bouquet B2 4 2 0 0 0 ... dipole D3 2 2 0 0 0 ... Ks,3 0 40 24 0 0 ... K X C3 2 38 24 0 0 ... The study of genus distributions began with [16]. Some of the early papers, such as [9] and [18], were devoted to calculating genus distributions for all the graphs in a recursively constructible sequence. Other early papers, such as [33] and [7], were concerned with statistical properties of the distribution. A solution to a genus distribution calculation problem can be either a formula or a polynomial-time algorithm. Lists of some previous papers on genus distributions have appeared in [10], [15], [17], and [29]. Papers published (or written) subsequently include the following: [4], [6], [5], [11], [12], [22], [23], [30], and [31]. Graph amalgamations and bar-amalgamations In general, amalgamating two graphs means identifying a subgraph in one of them to an isomorphic subgraph in the other. Figures 2 and 3 illustrate vertex-amalgamation and edge-amalgamation, respectively, which are the two simplest kinds of amalgamation of two graphs. A bar-amalgamation of two (disjoint) graphs G and H is obtained by joining a vertex u of G to a vertex v of H with a new edge. It is denoted here by G * H. Figure 4 shows a bar-amalgamation. Proposition 1.1 ([16]). Let G and H be (disjoint) connected graphs, and let u and v be vertices of G and H, respectively. Then Ydist(G * H) = deg(u) • deg(v) • Ydist(G) o Ydist(H) where o means the operation of convolution on two sequences. Seeking a useful algorithm The objective herein is to derive a quadratic-time algorithm for calculating the genus distribution of any 3-regular Halin graph. The focus is not merely on proving the existence of such an algorithm, but on developing an algorithm that can by executed (albeit tediously) by hand for graphs with 10-20 vertices and rather quickly by a computer for graphs with a significantly larger number of vertices. The terminology used here is consistent with [19] and [1]. For additional background (with some terminological differences), see [3], [28], or [37]. All of our graph embeddings here are cellular and orientable. A graph is taken to be connected, unless one can infer otherwise from the immediate context. Here we refer to a face-boundary walk as an fb-walk. Thanks to Imran Khan for creating the genus-distribution computer program (based on the Heffter-Edmonds algorithm) used in the course of this research. 2 Known results concerning genus distributions Although calculating the maximum genus Ymax (G) of a graph G is possible in polynomial time [8], calculating the minimum genus Ymin(G) is NP-hard [36], and calculating the genus distribution Ydist(G) is clearly at least as hard as calculating the minimum genus. Accordingly, rather few genus distributions are known. A survey of genus distributions, including average genus, is given by [10]. The most familiar such kinds of ladder graphs whose genus distribution formulas are known are as follows: closed-end ladders [9] (derived 1984) See Figure 5. circular ladders and Möbius ladders [26] See Figure 6. Ringel ladders [35] See Figure 7. By systematic use of iterated amalgamations [29] of double-edge-rooted graphs, self-edge-amalgamations [30], and edge-addition surgery [11], the calculation of formulas for these ladder graphs has been substantially simplified. Moreover, these recently developed techniques have produced quadratic-time algorithms for various generalizations of ladders, in which arbitrary graphs of known partitioned genus distribution (see §4) lie between the rungs. Figure 5: The closed-end ladder L4. Figure 6: Circular ladder CL4; and Möbius ladder ML4. » ^-1 »-1 »-( 1-1 4 »-' Figure 7: Ringel ladder RL4. A recent paper [13] presents a quadratic-time algorithm for the calculation of the genus distribution of any 3-regular outerplanar graph (see Figure 8). It uses a post-order traversal (see §3) and edge-amalgamations [29]. A subsequent paper [30] uses vertex-amalgamations [17] to derive a quadratic-time algorithm for 4-regular outerplanar graphs. Whereas outerplanar graphs are of tree-width 2, Halin graphs are of tree-width 3 (see [2]), which is intuitively a reason for anticipating the necessity for a more complicated analysis. Restrictions to 3-regularity or 4-regularity generally simplify the analysis of a genus distribution problem. Some genus-distribution deriviations use a formula of Jackson [21] based on the theory of group representations. Bouquets, which are graphs with a single vertex and a number of self-loops (see Figure 10) were the first class to be so derived [18] . Another such class is dipoles, which are graphs with two vertices and a number of edges joining them (see Figure 11). Their genus distributions are given by [32] and [24]. Yet another is fans, which are graphs obtained by joining a path graph to a single new vertex (see Figure 12). Their genus distributions were derived by [6]. I 0 0 Figure 11: Dipoles D1, D2, and D3. Figure 12: Fans F3 and F5. 3 Quadrangulating a plane Halin graph In deriving the genus distribution of Halin graphs, the critical problem was to invent a new form of decomposition of a plane Halin graph into "atomic" fragments whose genus distributions are known, a new form of amalgamation, and an order of reassembly that reconstructs the Halin graph from the atomic fragments. In this section, we concentrate on the decomposition and the reassembly. Taking the inscribed tree of a Halin graph as a spanning tree, an edge of a Halin graph is a tree-edge if it lies in the inscribed tree and a cycle-edge if it lies on the outer cycle. A leaf-edge is a tree-edge that is incident at a vertex of the outer cycle. A vertex is called a cycle vertex if it lies on the outer cycle, or an interior tree-vertex otherwise. We regard the vertices and the edges of the given plane Halin graph as black. We observe that since H is a Halin graph, there is exactly one cycle edge on each polygonal face of the plane embedding. The decomposition is a 4-step process. Step 1. In each cycle edge of the Halin graph, insert a red midpoint. This is illustrated in Figure 13. Step 2. Join each red vertex v to all of the non-leaf vertices on the boundary of the face in whose boundary v lies, as illustrated in Figure 14. Proposition 3.1. The red and black edges together triangulate the region inside the exterior cycle of a plane Halin graph G. Proof. The black edges create a set of polygons (whose number equals the cycle rank ß(G)). Each of these polygons is triangulated by the red edges. □ Proposition 3.2. Every black tree edge lies on two of the triangles formed by Steps 1 and 2. Proof. Every tree edge lies on two of the polygonal faces of the plane Halin graph (by the Jordan curve theorem). In each of those polygonal faces, it lies on one and only one of the Step 3a. For each black tree edge, we pair the two incident triangles into a quadrangle. Step 3b. We assign (unseen) colors blue, green, and brown to the tree edges, so as to form a proper edge 3-coloring. This is possible because any tree of maximum degree 3 is edge-3-colorable (via greedy algorithm). Step 3c. We visibly color each quadrangle with the unseen color of the tree edge that bisects it. The coloring of the quadrangles is a proper 3-coloring of the part of the plane inside the exterior cycle of the Halin graph, because of the way it is induced by the proper 3-edge-coloring of the tree. (This property will not be used, but it is interesting nonetheless.) Step 4. Separate the quadrangulated map into quadrangles, and label the interior tree-vertices. triangles. □ Figure 15: Quadrangulation of a plane Halin graph. Figure 16: Separated quadrangles of a plane Halin graph. Reassembling a Halin graph from its quadrangles: a puzzle The success of our method of calculation the genus distribution in the subsequent sections depends on our ability to reassemble the plane Halin graph from its separated quadrangles in a manner consistent with a puzzle now to be described. The genus distributions of the quadrangular fragments is known, and it will be shown that we can calculate the genus distribution of any graph that can be constructed from quadrangular fragments, according to the rules of this puzzle. After giving the rules for this puzzle, we consider the outcome of three attempts at its solution. Quadrangulation puzzle for a plane cubic Halin graph H ^ S0 1. Each quadrangle Q is regarded as an initial fragment. 2. An RR-path on a fragment boundary is a 2-path with two red edges, from a red vertex through a black vertex to another red vertex. 3. Initially, all RR-paths are said to be live. 4. A legal move is initiated by choosing a vertex v such that v is previously unchosen, at least one fragment at v is a quadrangle, and all three RR-paths through v are live RR-paths. If these three conditions are satisfied, then the three fragments that meet at v are merged into a single (larger) fragment. If there is more than one live RR-path on the boundary of the merged fragment, then all but one of the live RR-paths are deemed to be dead. 5. You LOSE if you run out of legal moves before the map is fully reassembled. This happens whenever there occurs an unmerged vertex w such that either there is a dead RR-path through w, or none of the fragments meeting at w is a quadrangle. 6. You WIN the game by reassembling the plane map. Attempt 1. Start with a merger at v. There are three live RR-paths on the boundary of the merged fragment. You LOSE, because RR-paths through two of the unmerged vertices u,w,x become dead. live RR-path Figure 17: Attempt #1. Attempt 2. First choose u and then choose v. There are two live RR-paths on the boundary of the merged fragment. You LOSE, because the RR-path through one of the unmerged vertices w, x becomes dead. Attempt 3. Start with u, w, y, z. You LOSE, since after there is a merger at v or x, there Solution: post-order traversal The post-order for the vertices of a plane tree is the order produced when one traces the boundary of the only region and calls out the name of a vertex only the last time it is visited. For the tree in Figure 20, the post-order is z, y, x, u, v, w. Figure 20: Post-order traversal. Solution for the quadrangulation puzzle 1. As a root for the inscribed tree of the Halin graph, choose any leaf-vertex. (Must be a leaf to win.) 2. Choose vertices in the order in which they occur on a post-order traversal of the tree. SOLUTION to puzzle in Figure 16: post-order as shown in Figure 20. z y x u v w \ v x ^r z W v x W z Figure 21: Solving the puzzle with post-order traversal. Theorem 3.3. Using the post-order of the interior tree-vertices as the order of merger solves the quadrangle puzzle for any plane cubic Halin graph. Proof. When the post-order is used, every RR-path through every vertex that follows the vertices of the fragment remains live. It also ensures that there is at least one quadrangle incident on each of those subsequent vertices. □ REMARK Quadrangulation and using the post-order solves the generalized puzzle for any Halin graph. The generalized algorithm is not presented only because its details are far lengthier than for the 3-regular case. 4 Partials and productions for Halin graphs When a graph G has one or more of its vertices or edges designated as roots, its genus distribution can be partitioned according to the ways in which face-boundary walks are incident on the roots. The components of such partitions are called partials. A surface-by-surface inventory of the values of the partials is called a partitioned genus distribution. Such partitioning has been a crucial step in most of the calculations of genus distributions. Here is a general paradigm for calculating of the genus distribution of the graphs in a given graph family F by various kinds of graph amalgamation. The tricky part is that all of these requirements must be satisfied in coordination with the others. • Prescribe a set A of rooted graphs as atomic fragments and a set M of merging operations, such that every member of F can be constructed by iterative application of the merging operations to the atomic fragments. We denote the closure of A under M as A. Thus, F C A. • A procedure is designed to determine, from any graph G in A, the sequence of application of operations from M to atomic fragments and to others constructed earlier in the sequence, by which graph G can be obtained. • An appropriate set of partials is developed for the rooted graphs in A. • For each operation ^ e M there is to be developed a set of rules, called productions, is developed, that prescribe the values of the partials of any graph in A from the values of the partials for the fragments that contribute to its construction under the operation Example 4.1. For the closed-end ladders and for the other kinds as well, the atomic fragments are doubly edge-rooted cycle graphs. The only operation for closed-end ladders is edge-amalgamation, and the order of application is linear. For the circular ladders and the Mobius ladders, there is an additional operation of self-edge-amalgamation, to be applied last. For the Ringel ladders, the additional operation is edge-addition, to be applied last. Example 4.2. For the cobblestone walks (see [9]), the atomic fragments are doubly vertex-rooted cycle graphs. The only operation is vertex-amalgamation. The order of application is linear. Example 4.3. For the 3-regular outerplanar graphs [12], the atomic fragments are doubly edge-rooted cycles. The operations are edge-amalgamation and root-popping on a singly edge-rooted graph. The order of operations is the post-order of a tree. For the 4-regular outerplanar graphs, the atomic fragments are doubly vertex-rooted cycles. The operations are vertex-amalgamation and root-popping. The order of operations is again the post-order of a tree. Atomic fragments and merging operations for Halin graphs The atomic fragments for constructing cubic Halin graphs are the quadrangular fragments obtained as in §3. We regard them here as doubly vertex-rooted. We denote this set of atomic fragments by AH . The only operation is merging three fragments at an interior vertex of the tree, in such a manner that either there is a surviving RR-path through the vertex of the fragment that is last (among the vertices of the fragment) in the post-order, or the Halin graph is fully reassembled. Order of mergers for Halin graphs The order of mergers of fragments is according to the post-order of the tree. Partials for cubic Halin graphs For a doubly vertex-rooted cubic Halin graph (G, u, v), with the roots u and v inserted at the midpoints of adjacent edges, we split gi(G) into six partials. Here is what they count: dd' Each of the roots u and v lies on two distinct fb-walks. One and only one of these fb-walks traverses both roots. dd'' Each of the roots u and v lies on two distinct fb-walks. Both of these fb-walks traverse both roots. ds' Root u lies on two distinct fb-walks. One of these fb-walks traverses root v twice. sd' Root v lies on two distinct fb-walks. One of these fb-walks traverses root u twice. ss1 A single fb-walks traverses roots u and v twice. The occurrences of each root are consecutive. ss2 A single fb-walks traverses roots u and v twice. The occurrences of the two roots alternate. These configurations are illustrated in Figure 22. dd' dd" ds' sd' ss1 ss2 Figure 22: The 6 double-rooted partials for a 3-way pie-merge. Proposition 4.1. Let G be any graph that is homeomorphic to a cubic graph, and let its vertex roots u and v be 2-valent endpoints of a pair of edges that are adjacent at a 3-valent vertex. Then the six partials dd', dd", ds', sd!, ss1, and ss2 completely partition the genus distribution of G. Proof. In every embedding of G, since u and v lie on a pair of edges that are adjacent at a 3-valent vertex, there is necessarily an fb-walk on which both of them occur. Thus, if both roots lie on two different fb-walks, dd' and dd'' are the only possibilities. If one lies on two different fb-walks and the other on only one fb-walk, then ds' and sd' are the only possibilities. If both roots occur twice on the same fb-walk, then either (ss1) the occurrences of each root are consecutive, or (ss2) they alternate. □ Productions for cubic Halin graphs For cubic Halin graphs, we merge three graphs at a time, exactly as for the puzzle, so that one of them is a quadrangle Q = K4 - e, with its two roots inserted at the midpoints of the two quadrangle boundary edges that meet at the vertex to be merged. Envisioning this configuration at a small pie cut into three slices, we call the a 3-way n-merge. It is illustrated in Figure 23. Figure 23: A 3-way n-merge ((A, r, s), (B,t,r'), (Q, s',t')) ^ (X,y,z) at vertex v. Proposition 4.2. In a 3-way n-merge (A, B,Q) ^ X at vertex v, each rotation system p for X is consistent with exactly two rotation systems for fragment A and exactly two for fragment B. Proof. If rotation system p is consistent with a given rotation system pA of fragment A, then it is also consistent with the rotation system of A obtained from pA by reversing the rotation at v. A similar observation holds for fragment B. □ Suppose that p1,p2,... ,ps is a set of partials or subpartials for a genus distribution. A production for a given surgical operation that transforms either a graph embedding X ^ Si (or a tuple of graph embeddings) into a set of graph embeddings of the graph Y is a rule of this form: pj (X ) Cipfj (i)(Y ) + ••• + Ct pfs (i)(Y ) The left side is called the antecedent, and the right side is called the consequent. The meaning is that the operation transforms a single embedding of graph X of type pj on the surface Si into a set of embeddings of the graph Y, of which ck are of type pk on the surface Sf j(i), for each i, j, and k. A drawing is usually used as an aid in deriving the production and in proving its correctness. The names of the graphs and their roots can be suppressed when there is in context no ambiguity. Example 4.4. One of the productions for the n-merge of doubly vertex-rooted graphs (A, r, s), (B, t, r') and (Q, s', t') into (X, y, z) is ddi (A, r, s) * ddj(B, t, r') —— 2d,d!i+j (X,y, z) + 2ss2i+j+1 (X,y,z) It means that a type-dd' embedding (A, r, s) — s' and a type-dd'' embedding (B, t, r') — S j combine into two type-dd' embeddings (X, y, z) — Si+j and two type-ss2 embeddings (X, y, z) — Si+j+1. The relevant drawing is shown in Figure 24. Theorem 4.3. The following 36 productions are a complete set of rules for calculating the genus distribution of the graph that results from a n-merge of three graphs in AH . 1. dd'i * ddj — - ddi+j + 2dd'i'+j+i + . 2. dd'i * ddj — - 2ddi+j + 2ss2+j+1. 3. dd'i * ds'j — ->• 2ddi+j + 2ss2+j+1. 4. dd'i * sd'j — 2sdi+j + 2ss1+j+1. 5. dd'i * sj — -)■ 4sd'i+j. 6. dd'i *ssi — -)■ 2dsi+j + 2sdi+j. 7. dd'i' * ddj — — 2ddi+j + 2ssi+j+1. 8. dd'i' * ddj — - 4ddi'+j. 9. dd'i' * ds'j — - 4ds'i+j. 10. dd'i' * sd'j — - 4sdi+j. 11. dd'i' * ss! — - 4ss1+j. 12. dd'i' *ssi — - 2ddi+j-1 + 2ss2+j. 13. dsi * ddj — ->• 2dsi+j + 2ss1+j+1. 14. dsi * ddj' — - 4ds'i+j. 15. dsi * dsj — 4ds'i' j. 16. dsi * sdj — "+4ss1+j. 17. dsi * ss! -*4ss1+j. 18. dsi *ssi — 2dsi+j-1 +2ss1+j. 19. sd'i * ddj — ->• 2ddi+j + 2ss2+j+1. 2 20. sdj * ddj — - 4sdj+j. 21. sdj * ds j — ->• 2ddj 1 + 2ss2+j. 22. sdj * sdj — 4sdj+j. 23. sdj * ssj — ->■ 4sdj j 1. 24. sdj * ss2 — -)■ 2ddj j 1 + 2ss2+j. 25. ss1 * ddj — - 4dsj+j. 26. ss1 * ddj — - 4ss1+j. 27. ss1 * dsj - - 4dsj 1 28. ss1 * sdj — - 4ss1+j. 29. ss1 * ssi — - 4ss1+j- 1. 30. ss1 * ss2 — - 4dsj j 1 31. ss2 * ddj — - 2dsj+j + 2sdj+j. 32. 2 ss2 * ddj — - 2ddj+j -1 + 2ss2+j. 33. 2 ss2 * dsj - - 2ddj+j. 1 + 2ss2+j. 34. 2 ss2 * sdj - - M+j- 1 + 2ss1+j. 35. 2 ss2 * ssj — - 4sdj+j- 1 36. 2 ss2 * ss2 — - 2ddj+j- 1 + ddj+j-2 + ssi+j-1. Proof. The correctness of each of these productions is a matter of recombining the strands as prescribed by the n-merge. The 36 figures corresponding to these productions are given by [14]. □ Computational Complexity Theorem 4.4. For |VA| = k and |VB | = m, there is an O(km)-time algorithm for calculating the partitioned genus distribution of the resulting graph X of a 3-way pie-merge (A, B, Q) - X of graphs whose maximum degree is 3. Proof. The number of non-zero partials of a cubic graph G with p vertices is in O(p), since the maximum genus cannot exceed ß(G)/2. For each non-zero-valued partial of A and each non-zero-valued partial of B, only one production is applied, and the time for the application of a single production is in O (1). □ Corollary 4.5. The post-order traversal using the 36 productions corresponding to the six partials yields an O(n2) algorithm for the genus distribution of a cubic Halin graph with n vertices. Proof. Let H have quadrangular fragments Q1,...,Qf of respective cardinalities q1,..., qf. The number of non-zero-valued partials in the n-merge of a k-vertex fragment A with an m-vertex fragment B and a quadrangular fragment Qj is at most a constant multiple of k + m. Since each pair of initial quadrangular fragments is merged only once during the reassembly of the Halin graph, it follows that the number of steps is at most a constant multiple of the sum where q' is the number of non-zero partials of the quadrangular fragment Q'. However, qqj < (qi + q2 +-----+ q/)2 The conclusion follows. □ 5 Sample Calculation In this section, we show the work needed to calculate the genus distribution of the Halin graph of Figure 1. Merger at z Graph A (K4 - e): i dd' dd'' ds' sd' ssi 2 ss2 g' 0 2 0 0 0 0 0 2 1 0 0 0 0 0 2 2 Graph B (KA - e): i dd' dd'' ds' sd' ssi 2 ss2 g' 0 2 0 0 0 0 0 2 1 0 0 0 0 0 2 2 Merged Graph K4: Use Productions 1, 6, 31, and 36. i dd' dd'' ds' sd' ssi 2 ss2 g' 0 2 0 0 0 0 0 2 1 0 4 4 4 0 2 14 Merger at y Merged Graph K4: Just like the merger at z. i dd' dd'' ds' sd' ssi 2 ss2 g' 0 2 0 0 0 0 0 2 1 0 4 4 4 0 2 14 Merger at x Graph A (result from merger at z): i dd' dd'' ds' sd' ssi 2 ss2 g' 0 2 0 0 0 0 0 2 1 0 4 4 4 0 2 14 Graph B (result from merger at y): i dd' dd'' ds' sd' ss1 o ssi g' 0 2 0 0 0 0 0 2 1 0 4 4 4 0 2 14 Merged Graph: Use 25 productions (all those without the partial ss1). i dd' dd'' ds' sd' ss1 o g' 0 2 0 0 0 0 0 2 1 40 4 12 12 0 2 70 2 0 16 48 48 32 40 184 Merger at u Merged Graph: K4: Just like the merger at z. i dd' dd'' ds' sd' ss1 o ss- g' 0 2 0 0 0 0 0 2 1 0 4 4 4 0 2 14 Merger at v Graph A (result from merger at x): i dd' dd'' ds' sd' ss1 o ssi g' 0 2 0 0 0 0 0 2 1 40 4 12 12 0 2 70 2 0 16 48 48 32 40 184 Graph B (result from merger at u): i dd' dd'' ds' sd' ss1 o ssi g' 0 2 0 0 0 0 0 2 1 0 4 4 4 0 2 14 Merged Graph: Use 30 productions. i dd' dd'' ds' sd' ss1 g' 0 2 0 0 0 0 0 2 1 112 4 28 12 0 2 158 2 544 96 544 352 80 112 1728 3 0 64 448 448 704 544 2208 Merger at w Graph A (result from merger at v): i dd' dd'/ ds'i sd'i ss1 ssi gi 0 2 0 0 0 0 0 2 1 112 4 28 12 0 2 158 2 544 96 544 352 80 112 1728 3 0 64 448 448 704 544 2208 Graph B (KA - e): i dd' dd/ dsi sd'i ssl 2 ss2 9i 0 2 0 0 0 0 0 2 1 0 0 0 0 0 2 2 Merged Graph: final result i ddi dd/ dsi sd'i ssl 2 ss2 di 0 2 0 0 0 0 0 2 1 144 4 60 4 0 2 214 2 1440 224 1632 224 56 144 3720 3 1024 1088 4800 1088 1088 1440 10528 4 0 0 0 0 896 1024 1920 6 Conclusions We have demonstrated the usefulness of the paradigm given at the beginning of §4 in deriving a practical algorithm for the genus distribution of cubic Halin graphs. To be practical, in the sense intended here, the number of partials needed should be relatively small. To calculate the genus distribution of a family of graphs, under this paradigm, one first designs a recursive specification of that family, that is, a finite set of base graphs and a finite set of operations whose iterative application can construct any graph in the family. One then derives a set of production rules for obtaining the partitioned genus distribution of the result of the applying any operation from the partitioned genus distributions of the operands. There are problems whose general solution seems to require exponentially large effort, but which can be solved in polynomial-time for cases in which something is bounded. A familiar result in topological graph theory is that whereas Thomassen [36] proved that determining the minimum genus of a graph is NP-hard, Mohar [27] proved that for every possible orientable surface S/, there is a linear-time algorithm to decide whether a given graph is embeddable in that surface. Shortly after the presentation of this paper, the author derived, for any fixed treewidth and maximum degree, a quadratic-time algorithm [15] to calculate the genus distribution of any graph conforming to those bounds. This algorithm is less than practical, since the numbers of partials and productions increase exponentially with the treewidth and the maximum degree. References [1] L. W. Beineke, R. J. Wilson, J. L. Gross and T. W. Tucker, Editors, Topics in Topological Graph Theory, Cambridge Univ. Press, 2009. [2] H. L. Bodlaender, A partial k-arboretum of graphs with bounded treewidth, Theoretical Comp. Sci. 209 (1998), 1-45. [3] C. P. Bonnington and C. H. C. Little, The Foundations of Topological Graph Theory, Springer, 1995. [4] Y. Chen, Lower bounds for the average genus of a CF-graph, Electronic J. Combin. 17 (2010), #R150. [5] Y Chen, J. L. Gross and T. Mansour, Genus distributions of star-ladders, Discrete Math. (2012), to appear. [6] Y. Chen, T. Mansour and Q. Zou, Embedding distributions of generalized fan graphs, Canad. Math. Bull. (2011), online 31aug2011. [7] J. Chen and J. L. Gross, Limit points for average genus (I): 3-connected and 2-connected sim-plicial graphs, J. Combin. Theory (B) 55 (1992), 83-103. [8] M. Furst, J. L. Gross and L. A. McGeoch, Finding a maximum-genus imbedding, J. ACM 35 (1988), 523-534. [9] M. L. Furst, J. L. Gross and R. Statman, Genus distribution for two classes of graphs, J. Combin. Theory (B) 46 (1989), 22-36. [10] J. L. Gross, Distribution of embeddings, in: L. W. Beineke, R. J. Wilson, J. L. Gross and T. W. Tucker (eds.), Chapter 3 of Topics in Topological Graph Theory, Cambridge Univ. Press, 2009. [11] J. L. Gross, Genus distribution of graphs under surgery: adding edges and splitting vertices, New York J. Math. 16 (2010), 161-178. [12] J. L. Gross, Genus distribution of graph amalgamations: Self-pasting at root-vertices, Australasian J. Combin. 49 (2011), 19-38. [13] J. L. Gross, Genus distributions of cubic outerplanar graphs, J. of Graph Algorithms and Applications 15 (2011), 295-316. [14] J. L. Gross, Productions for 3-way n-merges, http://www.cs.columbia.edu/~gross/supplementary.html. [15] J. L. Gross, Embeddings of graphs of fixed treewidth and bounded degree, Abstract 1077-051655, Boston Meeting oftheAmer. Math. Soc. (Jan. 2012). [16] J. L. Gross and M. L. Furst, Hierarchy for imbedding-distribution invariants of a graph, J. Graph Theory 11 (1987), 205-220. [17] J. L. Gross, I. F. Khan and M. I. Poshni, Genus distribution of graph amalgamations: pasting at root-vertices, Ars Combinatoria 94 (2010), 33-53. [18] J. L. Gross, D. P. Robbins and T. W. Tucker, Genus distributions for bouquets of circles, J. Combin. Theory (B) 47 (1989), 292-306. [19] J. L. Gross and T. W. Tucker, Topological Graph Theory, Dover, 2001; (original edn. Wiley, 1987). [20] R. Halin, Über simpliziale Zerfallungen beliebiger (endlicher oder unendlicher) Graphen, Mathematische Annalen 156 (1964), 216-225. [21] D. M. Jackson, Counting cycles in permutations by group characters, with an application to a topological problem, Trans. Amer. Math. Soc. 299 (1987), 785-801. [22] I. F. Khan, M. I. Poshni and J. L. Gross, Genus distribution of graph amalgamations at roots of higher degree, Ars Math. Contemp. 3 (2010), 121-138. [23] I. F. Khan, M. I. Poshni, and J. L. Gross, Genus distribution of P3 x Pn, Discrete Math. (2012), to appear. [24] J. H. Kwak and J. Lee, Genus polynomials of dipoles, KyungpookMath. J. 33 (1993), 115-125. [25] J. H. Kwak and J. Lee, Enumeration of graph embeddings, Discrete Math. 135 (1994), 129151. [26] L. A. McGeoch, Algorithms for two graph problems: computing maximum-genus imbedding and the two-server problem, PhD thesis, Carnegie-Mellon University, 1987. [27] B. Mohar, A linear time algorithm for embedding graphs in an arbitrary surface, SIAM J. Discrete Math. 12 (1999), 6-26. [28] B. Mohar and C. Thomassen, Graphs on Surfaces, Johns Hopkins Press, 2001. [29] M. I. Poshni, I. F. Khan and J. L. Gross, Genus distribution of edge-amalgamations, Ars Math. Contemporanea 3 (2010), 69-86. [30] M. I. Poshni, I. F. Khan and J. L. Gross, Genus distribution of 4-regular outerplanar graphs, Electronic J. Combin. 18 (2011), #P212. [31] M. I. Poshni, I. F. Khan and J. L. Gross, Genus distribution of graphs under self-edge-amalgamations, Ars Math. Contemp. 5 (2012), 127-148. [32] R. G. Rieper, The enumeration of graph embeddings, Ph.D. thesis, Western Michigan University, 1990. [33] S. Stahl, Region distributions of graph embeddings and Stirling numbers, Discrete Math. 82 (1990), 57-78. [34] S. Stahl, Region distributions of some small diameter graphs, Discrete Math. 89 (1991), 281299. [35] E. H. Tesar, Genus distribution of Ringel ladders, Discrete Math. 216 (2000), 235-252. [36] C. Thomassen, The graph genus problem is NP-complete, J. Algorithms 10 (1989), 568-576. [37] A. T. White, Graphs of Groups on Surfaces, North-Holland, 2001. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 57-68 Markov chain algorithms for generating sets uniformly at random Alberto Policriti Dipartimento di Matematica e Informatica, Università di Udine, Via delle Scienze, 206, 33100 Udine, Italy Alexandru I. Tomescu * Dipartimento di Matematica e Informatica, Universita di Udine, Via delle Scienze, 206, 33100 Udine, Italy Department of Computer Science, University of Bucharest, Str. Academiei, 14, 010014 Bucharest, Romania Received 30 November 2011, accepted 11 April 2012, published online 1 June 2012 Abstract In this paper we tackle the problem of generating uniformly at random two representative types of sets with n elements: transitive sets and weakly transitive sets, that is, transitive sets with atoms. A set is said to be transitive if any of its elements is also a subset of it; a set is weakly transitive if any of its elements, unless disjoint from it, is also a subset of it. We interpret such sets as (weakly) extensional acyclic digraphs—that is, acyclic digraphs whose (non-sink) vertices have pairwise different out-neighborhoods—and employ a Markov chain technique already given for acyclic digraphs. We thus propose Markov chain-based algorithms for generating uniformly at random (weakly) extensional acyclic digraphs with a given number of vertices. The Markov chain is then refined to generate such digraphs which are also simply connected, and digraphs in which the number of arcs is fixed. Keywords: Extensional digraph, transitive set, set theory, random generation, Markov chain. Math. Subj. Class.: 03E75, 05C81, 05C20 * Corresponding author. E-mail addresses: alberto.policriti@uniud.it (Alberto Policriti), alexandru.tomescu@uniud.it (Alexandru I. Tomescu) 1 Introduction Sets are basic mathematical and computational objects, and for this reason one is sometimes interested—in order to perform tests and evaluate benchmarks, collect statistical data, (dis)prove conjectures, etc.—in generating uniformly at random a set of given size. In this paper we tackle this problem by building on results originally given in the context of graphs. The sets we consider belong to the standard Zermelo-Fraenkel set theory and thus contain only other sets as elements. We focus on two archetypal sets: transitive sets and weakly transitive sets, that is, sets s whose elements are also subsets of s, and sets s whose elements, unless disjoint from s, are also subsets of s. A weakly transitive set s can be simply seen as a transitive set with atoms (or ur-elements), the atoms of s constituting the collection of elements of s disjoint from s. Zermelo-Fraenkel sets stand at the basis of programming languages such as SETL [15], or the more recent {log} [3] and CLP(SET) [5], and of the automatic proof-verifier Referee [14]. Since sets are nested structures, they are best represented by digraphs: vertices will stand for sets, while the arc relation will correspond to the inverse of the membership relation between them (^ = 3). Such digraphs are acyclic, since membership is well-founded, and weakly extensional, in the sense that distinct non-sink vertices have distinct collections of out-neighbors. This digraph interpretation was exploited in [13] to give a recurrence relation for the number of weakly transitive sets with n elements, generalizing the result of [12] for transitive sets with n elements. Under this graph-theoretic interpretation, we show in this paper how a Markov chain based procedure for generating acyclic digraphs, first introduced in [9], can be transferred to our set-theoretic universe. This Markov chain algorithm was already modified in [10] to generate simply connected acyclic digraphs. The random generation of elements from a particular class of acyclic digraphs modeling Bayesian networks was proposed in [6]. Finally, the same approach was used in [2] to generate deterministic acyclic automata. Each of these examples can be seen as a less basic case than the one tackled here. The idea behind this Markov chain technique is to start with an arbitrary weakly exten-sional acyclic digraph (w.e.a. digraph, for short) on n vertices and apply a certain number T of random local transformations which preserve weak extensionality and acyclicity. The uniformity of the resulting distribution is basically proved by showing that any w.e.a. digraph on n vertices can be thus transformed into any other w.e.a. digraph on n vertices. Like in the acyclic digraph case, we argue that the transformation rules are symmetric and always allow reaching a specific digraph among the collection of w.e.a. digraphs with n vertices. In our case, however, the most natural target digraph for this purpose turns out to be an acyclic tournament on n vertices, that is, the digraph whose interpretation in the universe of sets is the von Neumann numeral of n, the unique transitive set with n elements well-ordered by the membership relation. We prove here only 'correctness' and defer to future work computational aspects such as estimations for the choice of T or an analysis of the mixing time of the Markov chain [8]. As next research steps we also mention the random generation of hypersets, objects for which the acyclicity requirement is dropped, and weak extensionality is accordingly strengthened by an equality criterion based on the notion of bisimulation [1]. The paper is organized as follows. In Section 2 we give some notation and formally introduce the above-mentioned notions. In Section 3 we put forward a Markov chain for generating weakly extensional acyclic digraphs, while in Section 4 we propose a Markov chain for generating simply connected weakly extensional acyclic digraphs. The latter can be easily adapted to generate extensional acyclic digraphs. Finally, in Section 5, we present a Markov chain for generating w.e.a. digraphs on n vertices and m arcs, 0 < m < ("), which can also serve to generate acyclic digraphs with a given number of vertices and of arcs. 2 Notation and preliminaries We consider only finite simple digraphs, that is, without parallel arcs or self-loops. Given a digraph D, we denote by V(D) its vertex set and by E(D) the set of its arcs. For any v e V (D), N + (v) stands for the set {u e V (D) | (v,u) e E (D)}, which is called the out-neighborhood of v in D. Similarly, N -(v) = {u e V (D) | (u, v) e E (D)} is the in-neighborhood of v. We will use d+(v) = |N+(v)| and d-(v) = |N-(v)|. A vertex v e V (D) such that d+(v) = 0 will be called a sink; if d-(v) = 0, it will be called a source; we denote by I(D) the set of the sinks of D. A digraph D is said to be simply connected if the underlying (undirected) graph of D is connected. If (i, j ) e E (D), we will employ D - (i, j) as a shorthand for the digraph obtained from D by removing the arc ( i, j ). Analogously, D + (i, j) is the digraph obtained from D by the addition of the arc (i, j). Definition 2.1. A digraph D is said to be • extensional, if for any distinct u,v e V (D) it holds that N+ (u) = N+(v); • weakly extensional, if for any distinct vertices u,v e V (D) \ I (D), it holds that N+(u) = N +(v). If u, v are distinct vertices of a digraph D having N +(u) = N +(v), we say that u and v collide. Note that this is not the case if D is acyclic and there is a directed path from u to v. In particular, in an acyclic tournament there are no collisions (a tournament is a digraph D such that for any distinct u, v e V (D) either (u, v) e E (D) or (v, u) e E(D) holds, but not both). Moreover, any e.a. digraph is simply connected. Under the Zermelo-Fraenkel axioms, each set is uniquely characterized by its elements (Extensionality Axiom), and the membership relation is well-founded (Foundation Axiom). The standard universe of sets is von Neumann's cumulative hierarchy, whose subset of hereditarily finite sets is defined as the union, over all natural numbers i, of Vj, where V0 = 0, each level Vi+1 is P(Vj), and P(-) stands for the power-set operator. For example, Vi = {0}, V2 = {0, {0}}, V3 = {0, {0}, {{0}}, {0, {0}}}. To faithfully represent a set x as a digraph, one considers the digraph Dx defined as Dx = (x, {(u, v) | u, v € x, v € u}). See Figure 1 for an example. Since the membership relation between sets is well-founded, such digraphs are acyclic. Moreover, if x is transitive, then Dx is also extensional, while if x is weakly transitive, then Dx is weakly extensional. To see this, observe that if distinct non-sink vertices have the same out-neighborhood, then from the (weak) transitivity assumption they correspond to the same set, contradicting the definition of Dx; see also [13]. Given n > 1, we denote by Wn the set of all w.e.a. digraphs with vertex set {1,..., n}, while Wn denotes its subset of simply connected w.e.a. digraphs. Analogously, Wn,m denotes the set of all w.e.a. digraphs with vertex set {1,..., n} and m arcs. Definition 2.2. A discrete time finite stochastic process is a sequence X = (Xt : t € N), where Xt are S-valued random variables and S is a finite set, called the state space of X. We say that X is a Markov chain if Vt € N, Pr(Xt+i = st+i | Xt = st,...,X o = so) = Pr(Xm = sm | Xt = st). Moreover, a Markov chain X is said to be time-homogeneous if Vs, s' € S, 3pss,, Vt € N, Pr(Xt+i = s | Xt = s') = pss,. Definition 2.3. A time-homogeneous Markov chain over the state space S is said to be: • irreducible iff Vs, s' € S, 3t € N, Pr(Xt = s' | X0 = s) > 0; • aperiodic iff Vs € S, gcd{t € N | Pr(Xt = s | X0 = s) > 0} = 1; • symmetric iff Vs, s' € S, pss/ = ps/s. A well-known result (see, e.g., [8]) states that any finite, irreducible, aperiodic and symmetric time-homogeneous Markov chain converges toward the uniform distribution on its state space. Therefore, all the Markov chains presented here will be shown to satisfy these three properties. 3 A Markov chain algorithm for generating w.e.a. digraphs Let M be a Markov chain over Wn, defined in Figure 2. Observe that M differs from the Markov chain of [9] for generating arbitrary acyclic digraphs in the fact that arc deletions and additions are done only provided that the resulting digraph is w.e.a. Notice that for any t € N and any two distinct states s, s' € Wn, Pr(Xt+1 = s | Xt = s') > 0 if and only if Pr(Xt+1 = s' | Xt = s) > 0. To be more precise, the probability of passing from a state s € Wn to any other state s' = s is either 0 or 1/n2, hence M is symmetric. Moreover, for every s € Wn the probability of remaining in s at any t > 0 is positive, for example by choosing diagonal pairs (i, i), with i € {1,..., n}. Therefore, if M turns out to be irreducible, then it will also be aperiodic. The initial state of this Markov chain and of the ones given in the next section can be taken to be a linear digraph consisting of a directed path (n, n - 1,..., 1). The acyclicity of a digraph on n vertices and m arcs can be established by a depth-first visit, in time O(n + m). To test whether a digraph is (weakly) extensional, the algorithm in [4, Sec. 4] can be used, taking time O(n + m). Let Xt denote the state of the Markov chain at time t. Suppose a couple of integers (i, j) has been drawn uniformly at random from the set {1,. .., n} X {1,. .., n}. (Ti) if (i,j ) G E (Xt) and Xt - (i, j) is w.e., then Xm = Xt - (i, j) else Xt+i = Xt. (T2) if (i,j ) G E (Xt) and Xt + (i, j) is w.e.a., then Xt+i = Xt + (i, j) else Xt+i = Xt._ Figure 2: A Markov chain algorithm for generating w.e.a. digraphs. Lemma 3.1. Let D be a w.e.a. digraph with E(D) = 0. There exists an arc (u, v) G E (D) such that the digraph D — (u, v) is w.e.a. Proof. Observe first that there exists u G V (D) such that 0 = N+(u) C I (D). If this were not the case, then for all u in V(D) with N +(u) = 0, there would exist a vertex u' in D with N +(u') = 0 such that (u, u' ) G E (D). Since the same property holds for u' as well, and as the number of vertices of D is finite, we can find a finite directed cycle, contradicting hence the acyclicity of D. def Let now U (D) be the set of vertices of D with the above property, that is, U (D) = {u G V (D) | 0 = N +(u) C I (D)}. Let u0 G U (D) be a vertex of minimum out-degree, i.e., d+(u0) = min{d+(u) : u G U (D)}. Since N+(u0) = 0, let v0 be an element of N +(u0). The arc (u0,v0) can be removed and the resulting digraph remains w.e.a. Indeed, since u0 is among the vertices of minimum out-degree in U (D), in D — (u0, v0) it will either be a sink, or it will be the only vertex in U (D — (u0, v0)) with out-degree d+(u) — 1, hence having its out-neighborhood different from that of any other vertex of D — (u0,v0). □ Theorem 3.2 (Irreducibility of M). Let M be the Markov chain defined over the space Wn together with the transition rules T1 and T2. Given two distinct digraphs D and H in Wn, there exists in M a sequence of transitions D = D0 ^ D1 ^ • • • ^ Dp-1 ^ Dp = H, where p > 1 and Di G Wn, for all 0 < i < p. Such a sequence exists with length at most n2 — n. Proof. Since M is symmetric, it suffices to show that there exists a sequence of transitions from any given w.e.a. digraph D G Wn to a fixed element O in Wn. For our purpose here, we will choose O to be the unique totally disconnected digraph, that is, having E(O) = 0. From Lemma 3.1, we get that there exists an arc (u, v) G E(D) such that D — (u, v) is w.e.a. Using rule (T1), we can step from D to D — (u, v). Repeating the above argument a finite number of steps, we arrive at O. The number of transitions from D to O is at most n(n — 1)/2, and this is obtained when D is a tournament. □ 4 A Markov chain algorithm for generating e.a. digraphs Instead of generating e.a. digraphs, we place ourselves in a more general setting, that of generating simply connected w.e.a. digraphs. Afterwards, we will argue that, with minor changes, the proposed Markov chain can generate e.a. digraphs. Let Mc be the Markov chain over WC whose transitions between states are given in Figure 3. This Markov chain is adapted from [10], with the difference that arc deletions, additions, or reversals are done only if the resulting digraph is simply connected and w.e.a. This simple modification, however, requires a totally new and more involved proof of irreducibility. Just as in the previous section, the probability of passing from a state s G WC to any other state s' = s is either 0 or 1/n2, implying that Mc is symmetric. Likewise, the aperiodicity of Mc will be implied by its irreducibility and by the fact that for every state in WCc there is a positive probability to remain in that state (by choosing diagonal pairs (i, i)). Even if the two transition rules of Mc are not entirely specular, one can think of Mc as having three basic transitions: (1) removal of an arc, (2) reversal of an arc, (3) addition of an arc. According to this view, Mc is entirely symmetric. Let Xt denote the state of the Markov chain at time t. Suppose a couple of integers (i, j) has been drawn uniformly at random from the set {1,. .., n} X {1,. .., n}. (T1) if (i,j) G E(Xt) then (a) if Xt - (i, j) is simply connected and w.e., then Xt+i = Xt - (i, j), else (b) if Xt - (i, j) + (j, i) is w.e.a., then Xm = Xt - (i, j) + (j, i), (c) else Xt+i = Xt. (T2) if (i,j) GE(Xt), then (a) if Xt + (i, j) is w.e.a., then Xt+i = Xt + (i, j), (b) else Xt+i = Xt._ Figure 3: A Markov chain algorithm for generating simply connected w.e.a. digraphs. To show the irreducibility of the Markov chain Mc, it is useful to partition the vertices of an acyclic digraph according to the maximum length of a directed path to a sink of the digraph. Complying with standard set-theoretic notation, we will make use of the following notion. Definition 4.1. Given an acyclic digraph D, the rank of a vertex v G V (D) is recursively defined as rk(v) = 1 + max{rk(u) : (v, u) G E(D)}, where rk(v) =0 if v is a sink. Clearly, the following lemma holds. Lemma 4.2. Given an acyclic digraph D, if v,u G V (D) and rk(v) = rk(u), then N +(v) = N + (u) holds. Throughout the subsequent two proofs we employ the following notation: given an acyclic digraph D and a vertex v of D, R(v) d=f {u G V (D) | u = v and rk(u) < rk(v)}. Theorem 4.3 (Irreducibility of Mc). Let Mc be the Markov chain defined over the space Wn together with the transition rules Ti and T2. Given two distinct digraphs D and H in Wn, there exists in Mc a sequence of transitions D = D0 ^ Di ^ • • • ^ Dp-i ^ Dp = H, where p > 1 and Di e WCC, for all 0 < i < p. Such a sequence exists with length at most (3n2 — 7n + 4)/2. Proof. As before, first we will show that there exists a sequence of transitions from any given w.e.a. digraph D e Wn to an element T (D) in Wn, where T (D) is an acyclic tournament, with the additional property that whenever rk(v) > rk(u) in D, rk(v) > rk(u) also holds in T (D). Then, given any D and H in Wn, we will show that there is a sequence of transitions from T (D) to T (H ), completing hence the proof. To show the former claim, we proceed as follows. Pick a vertex v e V (D), in decreasing order of rank (when more vertices of the same maximum rank exist, pick an arbitrary one). Apply rule (T2) and add arcs from v to all the vertices u e R(v) \ N + (v), in decreasing order on the rank of the elements of R(v) \ N + (v). Note that this is possible, first of all, because the addition of an arc (v, u) does not create a cycle in the resulting digraph. Second, observe that the subdigraph of D induced by the vertices V (D) \ R(v) is an acyclic tournament. Therefore, an arc addition would create a collision only between v and a vertex w e R(v). This is however not the case, since after the first addition of such an arc, rk(v) becomes strictly greater than rk(w), for all w e R(v), and Lemma 4.2 guarantees the absence of collisions. Denote by T (D) the acyclic tournament obtained at the end of this process. Since for any vertex v we have added arcs only to those vertices of rank less than or equal to v, we also have that whenever rk(v) > rk(u) in D, the same holds in T (D).1 Passing on to the latter point, observe that for any w.e.a. digraph D, since T (D) is a tournament, there are no two distinct elements of the same rank in T (D), and thus {rk(v) : v e V (T (D))} = {0,..., n — 1}. Hence, to each digraph T (D) we can uniquely associate a linear order -) xi. Starting from position i = 0 all the way to i = n — 1, apply the following procedure. If yi = xj, (i < j < n — 1), then bring xj to position i by iteratively reversing the arcs (xj, xj-i), (xj, xj — 2), . ..,(xj, xi). The maximum number of transitions to pass from D to T (D) is (n) — (n — 1) = (n2 — 3n + 2)/2, number obtained when the underlying graph of D is a tree, thus having n — 1 edges. To pass from T (D) to T (H ), (n) transitions are required at most, when all the arcs of T (D) have to be reversed. Hence, to pass between two arbitrary D and H in Wn, we need at most (3n2 — 7n + 4)/2 transitions. □ 1 However, the converse in general does not hold. Let us denote by En the set of all e.a. digraphs with vertex set {1,..., n}. The Markov chain illustrated in Figure 3 can be transformed into an irreducible, aperiodic and symmetric Markov chain, Me, for the generation of digraphs from En. The transitions between two states in Me are given in Figure 4. The analogue of Theorem 4.3 holds for Me as well. Let Xt denote the state of the Markov chain at time t. Suppose a couple of integers (i,j) has been drawn uniformly at random from the set {1,. .., n} X {1, ..., n}. (T1) if (i,j) e E (Xt) then (a) if Xt - (i,j) is extensional, then Xt+i = Xt - (i,j), else (b) if Xt - (i, j) + (j, i) is e.a., then Xm = Xt - (i, j) + (j, i), (c) else Xt+i = Xt. (T2) if (i,j) e E (Xt), then (a) if Xt + (i, j) is e.a., then Xt+i = Xt + (i, j), (b) else Xt+i = Xt._ Figure 4: A Markov chain algorithm for generating e.a. digraphs. 5 A Markov chain algorithm for generating w.e.a. digraphs with a given number of arcs A Markov chain Ma for generating w.e.a. digraphs with vertex set {1,..., n} and m arcs is given in Figure 5. It will immediately follow from the proof of its irreducibility—Theorem 5.1 below—that Ma can also generate uniformly at random acyclic digraphs with a given number of vertices and a fixed number of arcs, by simply swapping two arcs if the resulting digraph is acyclic. Note that controlling the number of arcs was already considered in the literature: [9] proposed generating acyclic digraphs with a bounded number of arcs, or whose vertices have a bounded degree; [7] proposed generating acyclic digraphs with bounded induced width, a complexity measure arising from artificial intelligence. The probability of passing from a state s e Wn,m to any other state s' = s is either 0 or 1/n4, implying that Ma is symmetric. As previously, for any state s e Wn,m there is a positive probability to remain in s. Our next theorem shows that Ma is indeed irreducible. If m < n - 1, the initial state of the Markov chain can be a digraph whose arcs form a directed path of length m. Otherwise, the initial state can be a directed path (n, n-1,..., 1) together with m - (n - 1) arbitrary arcs of the form (i, j), where i > j. Theorem 5.1 (Irreducibility of Ma). The Markov chain Ma is irreducible. Proof. First, if m = 0, Wn,m consists only of the totally disconnected digraph. Assuming that m > 0, we show that any digraph D e Wn,m can be transformed, by transitions of Ma, into a digraph K (D) e Wn,m, satisfying the following two properties: i) for all v e V (D) such that rk(v) > 1 in K (D), it holds that N + (v) = R(v) in K(D); ii) there is only one v e V (D) such that rk(v) = 1 in K (D). Let Xt denote the state of Ma at time t. Suppose two pairs of integers (il, jj) and (i2, j2) have been drawn uniformly at random and independently from the set {1,..., n} x {1,..., n}. if (ii, ji) G E(Xt) and (i2, j2) G E(Xt), then if Xt - (il, ji) + (i2, j'2) is w.e.a., then Xm = Xt - (ii, ji) + (i2, j'2), else Xt+i = Xt._ Figure 5: A Markov chain algorithm for generating w.e.a. digraphs on n vertices and m arcs. To show this, we argue as in the proof of Theorem 4.3, paying particular attention to preserving m arcs at each intermediate step. Observe that if D fails to satisfy i) or ii), then it must own a vertex v such that N + (v) C R(v). Indeed, first note that N+(v) C R(v) holds for any vertex v, since D is acyclic. If D does not satisfy i) and v G V (D) is a vertex with rk(v) > 1 and N +(v) = R(v), then N + (v) C R(v) immediately follows. If D owns two distinct vertices v' and v" of rank 1, then v" G R(v') \ N + (v'). Therefore, as long as D fails to satisfy one of the above conditions i) or ii), apply the following transformation to it. Pick a vertex v G V (D) inclusion-maximal among the vertices of maximum rank having N+(v) C R( v), and consider all the elements u G R(v)\ N + (v), in decreasing order on rank. Take t G V (D) a vertex whose out-neighborhood is inclusion-minimal among the vertices of rank 1. Arguing as in the proof of Lemma 3.1, any arc (t, s) leading from t to an arbitrary sink s G V (D) can be removed without disrupting the weak extensionality of D. Replace arc (t, s) with arc (v, u) by a transition of Ma. This is possible, since, on the one hand, the addition of the arc (v, u) does not create a cycle in the resulting digraph. On the other hand, as before, the subdigraph of D induced by the vertices V (D) \ R(v) is an acyclic tournament. Therefore, one such arc addition can create a collision only between v and some vertex w g R(v). This is not the case, since after the first addition of an arc from v to a maximum rank vertex in R(v), rk(v) becomes strictly greater than rk(w), for all w G R(v), and Lemma 4.2 guarantees the absence of collisions. It should be clear that the above procedure stops after a finite number of steps, and that the final digraph satisfies conditions i) and ii). It remains to show that, given two digraphs D and H in Wn,m, there exists a sequence of transitions in Ma from K (D) to K (H ). To any acyclic digraph K whose vertices of positive rank have pairwise distinct ranks we can associate a partial order - 1,then N+(xj+1) = N+(xj) U{xj}. The arc (xi+1,xj) can be reversed, by the application of the transition of Ma on the arcs (xj+1, xž) and (xž, xj+1). Indeed, the resulting digraph K' remains acyclic; K' is also w.e. since, on the one hand, vertices xj, xj+1,..., xn-1 induce an acyclic tournament in K', by conditions i) and ii). On the other hand, any non-sink xj, 0 < j < i, is an out-neighbor of both xj and xj+1. Moreover, if rk(xj+1) was equal to 2 in K (D) (and hence rk (xj) = 1), then in K ' we may have N+(xj) = R(xj). However, it suffices to swap arcs out-going from xj+1, the unique vertex of rank 1 in K', to xj. The digraph obtained after these transformations satisfies conditions i) and ii), thus is equal to some K (D'); the vertices of K (D') have the same ranks as in K (D), with the exception of xj and xj+1 which have swapped ranks. When however rk(xj+1 ) = 1, we have that rk(xj) = 0. Since xj+1 is the unique vertex of rank 1, there must be an arc from xj+1 to a sink s (s can even be xj) which can be removed in order to add the arc (xj, xj+1). After this first arc swap, continue changing all arcs (xj+1, s) into (xj, s), with s an arbitrary sink. The resulting digraph K' satisfies conditions i) and ii) and is equal to some K (D'); moreover, the vertices of K (D') have the same ranks as in K (D), with the exception of xj and xj+1 which, as before, have swapped ranks. In order to transform K (D) into K (H ), apply the following procedure, starting from position i = 0 all the way to i = n — 1. If yj = xj, (i < j < n — 1), then bring xj to position i by iteratively reversing the arcs (xj, xj-1), (xj, xj-2), ... ,(xj, xj). Finally, change the out-going arcs of the unique vertex of rank 1 so that it has precisely the same out-neighborhood as it has in K (H ). □ Figure 6 illustrates the transitions indicated by the above proof in order to pass between two digraphs in W5,6. 6 Critical remarks and future work Although the Markov chains M, Mc and Me are similar to the Markov chains of [9, 10], the proofs of their irreducibility are different and more involved. In the case of M, the fixed element which can be reached by a chain of transitions from every element D of Wn is the same as in [9], namely the totally disconnected digraph. However, the arcs of D must be removed in a particular order, respecting the weak extensionality of D. Second, on the one hand, in [10] the fixed element is an arbitrary digraph having a path as underlying graph, which cannot be the case for Mc or Me since (weak) extensionality would be violated. On the other hand, our proof takes this fixed element to be an acyclic tournament on n vertices, ensuring that the proof proposed here can also show the irreducibility of (a slightly modified version of) the chain of [10]. Lastly, as already noted, the Markov chain Ma can be easily adapted to generate uniformly at random acyclic digraphs on a given number of labeled vertices and a given number of arcs, a result which we have not found in the literature. Given this dual usability of the Markov chains considered here, and the fact that the acyclic tournament on n vertices (that is, the digraph isomorphic to the von Neumann numeral of n) is a rich structure in which many types of digraphs can be embedded, it would be interesting to give a general characterization of the classes of digraphs whose elements can be generated uniformly at random by these Markov chains. We regard the generation of digraphs on n vertices, possibly having directed cycles, (a) (4, 5) « (3,5) (b) (2, 4) « (3, 4) (c) 2 ^ 4 ^ 5 ^ 1 ^ 3 (i, 5) « (5, 1) (i, 4) « (5, 4) (d) i ^ 2 ^ 4 ^ 5 ^ 3 (e) i ^ 2 ^ 4 ^ 3 ^ 5 (3, 5) « (5, 3) (3, 4) « (4, 3) (3, 2) « (5, 2) (3, i) « (4, i) (f) i ^ 2 ^ 3 ^ 4 ^ 5 Figure 6: The sequence of transitions of Ma that transforms D G W5,6 (Fig. (a)) into K (D) G W5,6 (Fig. (c)), and then into a digraph K(H ) G W5,6 (Fig. (f)) but deprived of distinct bisimilar vertices (the standard interpretation of a hyperset [1]) as the next natural step to take. It is interesting to study whether a Markov chain algorithm consisting of three basic operations: addition of an arc, removal of an arc, move of an arc, is irreducible for the class of such digraphs. As already mentioned in the introduction, the problem of analyzing the mixing times [8] of our proposed Markov chains remains open; this is important from a practical viewpoint. Finally, since a peculiarity of the sets treated in this paper is to be sets whose elements are themselves sets, it would be interesting to investigate what role our results can play in the theory of random sets ([11], elements taking as values subsets of some space), for example as a tool to generate such objects uniformly. Acknowledgements The authors are grateful to the anonymous referees for their useful suggestions and comments which greatly improved the original version of this paper. References [1] P. Aczel, Non-Well-Founded Sets, volume 14 of CSLI Lecture Notes, CSLI, Stanford, CA, 1988. [2] V. Carnino and S. De Felice, Random Generation of Deterministic Acyclic Automata Using Markov Chains, in: B. Bouchou-Markhoff, P. Caron, J.-M. Champarnaud and D. Maurel (eds.), Implementation and Application of Automata, volume 6807 of LNCS, Springer Berlin / Heidelberg, 2011, pp. 65-75. [3] A. Dovier, E. G. Omodeo, E. Pontelli and G. Rossi, {log}: A Language for Programming in Logic with Finite Sets, J. Log. Program. 28 (1996), 1-44. [4] A. Dovier, C. Piazza and A. Policriti, An efficient algorithm for computing bisimulation equivalence, Theor. Comput. Sci. 311 (2004), 221-256. [5] A. Dovier, C. Piazza, E. Pontelli and G. Rossi, Sets and constraint logic programming, ACM Trans. Program. Lang. Syst. 22 (2000), 861-931. [6] J. S. Ide and F. G. Cozman, Random Generation of Bayesian Networks, in: G. Bittencourt and G. L. Ramalho (eds.), 16th Brazilian Symposium on Artificial Intelligence, SBIA 2002, volume 2507 of LNAI, Springer Berlin / Heidelberg, 2002, pp. 366-376. [7] J. S. Ide, F. G. Cozman and F. T. Ramos, Generating random bayesian networks with constraints on induced width, in R. López de Mantaras and L. Saitta (eds.), ECAI, IOS Press, 2004, pp. 323-327. [8] D. A. Levin, Y. Peresand and E. Wilmer, Markov Chains and Mixing Times, AMS, Providence, 2009. [9] G. Melancon, I. Dutour, and M. Bousquet-Melou, Random generation of directed acyclic graphs, in J. Nesetril, M. Noy and O. Serra (eds.), Comb01, Euroconference on Combinatorics, Graph Theory and Applications, volume 10 of Electronic Notes in Discrete Mathematics, 2001, pp. 202-207. [10] G. Melancon and F. Philippe, Generating connected acyclic digraphs uniformly at random, Information Processing Letters, 90 (2004), 209-213. [11] I. Molchanov, Theory of Random Sets, Springer-Verlag London, 2005. [12] R. Peddicord, The number of full sets with n elements, Proc. Amer. Math. Soc 13 (1962), 825-828. [13] A. Policriti and A. I. Tomescu, Counting extensional acyclic digraphs, Information Processing Letters 111 (2011), 787-791. [14] J. T. Schwartz, D. Cantone and E. G. Omodeo, Computational Logic and Set Theory, Springer, 2011. [15] J. T. Schwartz, R. B. K. Dewar, E. Dubinsky and E. Schonberg, Programming with Sets: An Introduction to SETL, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1986. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 69-81 Complex parameterization of triangulations on oriented maps Mathieu Dutour SikiriC * Rudjer BoskoviC Institute, Bijenićka 54, 10000 Zagreb, Croatia Received 12 October 2011, accepted 9 April 2012, published online 1 June 2012 Abstract We consider here triangulations of oriented maps having a specified set S of vertices of degree different from 6 and some other vertices of degree 6. Such map can be described by specifying the relative positions between elements of S using Eisenstein integers. We first consider the case of 1 parameter, which corresponds to the Goldberg-Coxeter construction. Then we develop the general theory, the special case of positive curvature studied by Thurston and finally extend the theory to quadrangulations and some other cases. In the last section we expose application of parameterizations to the study of zigzags. Keywords: Maps, graphs, Groups, parameterizations. Math. Subj. Class.: 05C10, 57M20 1 Triangulations of oriented maps By a (jvi,..., vm}, k)-map we denote a map on an oriented surface with faces of size k, v vertices of degree i and "map" being "sphere" (genus g = 0), "torus" (g = 1) or "oriented map of genus g". For example, a ({v5 = 2, v7 = 2, v6}, 3)-torus denotes a triangulation with 2 vertices of degree 5, 7 and an unspecified number of vertices of degree 6. We will be mostly concerned with the description of (v, 3)-maps of genus g. Euler relation for them reads as £ vj(6 - j) = 12(1 - g). (1.1) j >3 The dual of ({v5 = 12, v6}, 3)-spheres are 3-regular plane graphs, whose faces are 5- or 6-gonal. Such graphs, named fullerenes, occur in chemistry [13] following the discovery of Buckminsterfullerene (also called truncated icosahedron, soccer ball) in 1985 [22]. *The author has been supported by the Croatian Ministry of Science, Education and Sport under contract 098-0982705-2707. E-mail address: mdsikir@irb.hr (Mathieu Dutour Sikiric) Formula (1.1) can be interpreted as a Gauss-Bonnet formula and 6 - j as the curvature of a vertex of degree j. A triangulation is said to be of positive curvature if all vertices have non-negative curvature. This implies that the possible vertex degrees belong to {3,4,5,6} and the v-vector satisfies 3v3 + 2v4 + v5 = 12 with v6 unspecified. All 19 possibilities for (v3, v4, v5) are given below: (0,0,12) (0,1,10) (0, 2, 8) (0, 3, 6) (0,4, 4) (0, 5, 2) (0, 6,0) (1,0, 9) (1,1, 7) (1, 2, 5) (1, 3, 3) (1,4,1) (2,0, 6) (2,1,4) (2, 2, 2) (2, 3,0) (3,0, 3) (3,1,1) (4,0, 0) For the enumeration of 3-regular plane graphs with a specified face vector (p^, i.e. number Pi of faces of size i the program cpf by T. Harmuth [2] is very efficient. Another little known program by the same author is cgf [19], which can enumerate 3-regular oriented maps of specified genus and face vector. The corresponding program for 4-regular plane graphs is enu by O. Heidemeier [2]. All above mentioned programs are available from [1] and give by duality triangulations and quadrangulations. The symmetry groups of fullerenes and other plane graphs of positive curvature were determined in [13, 5, 12, 7, 8, 6]. For a given group G of symmetry of a map, denote by Rot(G) the subgroup of index 1 or 2 of G formed by the orientation preserving transformation. The class of a group G is the set of groups G' having Rot(G') = G. In Table 1 we give the possible groups of (v, 3)-spheres of positive curvature by their class Rot(G), where we used Schoenflies nomenclature for point groups. For any class the number of vertices of positive curvature is finite and the number of vertices of degree 6 is unspecified. Since there is essentially only one 6-regular plane triangulation, one sees that the positions of the vertices of positive curvature allow to define the map. We want to encode the positions by complex Eisenstein numbers z G Z[w] with w = ein/3. In Section 2 we describe the simple cases of 1 or 2 parameters. The case of 1 parameter corresponds to the Goldberg-Coxeter construction [9]. In Section 3 we first explain the general theory of complex parameterization of (v, 3)-maps on oriented surfaces. Then we explain Thurston's theory [28] which gives stronger results for the case of spheres of positive curvature. Finally we explain the extension to (v, 4)-maps, self-dual spheres and (v, 6)-spheres. Applications to zigzags are considered in Section 4. A very basic application of parameterization is for generating maps efficiently provided that the number of parameters is not too high. Another application considered in [11] is for eigenvalue estimation where it was proved that for any interval [a, b] C [-3,3] there is a finite number of graphs of positive curvature having no eigenvalue in I. We choose to emphasize Eisenstein parameter description but it is of course possible to consider descriptions by integral parameters. This is done in [16,15] for fullerenes and this allows to write parameterizations for each group, not just rotation subgroup. Another real parameter descriptions, by so called dihedral angles, is developed in [26], but it is more suited for describing manifolds than graphs. 2 One and two parameter constructions 2.1 The Goldberg-Coxeter construction The Goldberg-Coxeter construction takes a (v, 3)-map M, two integers k, I > 0 and returns another ({v, v6}, 3)-map GCfcjI (M) of the same genus by adding triangles and vertices of ({v3 = 4, ve}, 3)-spheres class all group # param ({v5 = 12, ve}, 3)-spheres D2 D2, D2d, D2h 2 class all group # param T T, Td 1 Ci C1, Cs, Ci 10 C2 C2, C2h, C2v 6 ({v4 = 6, ve}, 3)-spheres C3 C3, C3h, C3v 4 class all group # param D2 D2, D2h, D2d 4 Ci C1, Cs, Ci 4 D3 D3, D3h, D3d 3 C2 C2, C2v, C2h 3 D5 D5, D5h, D5d 2 D2 D2, D2d, D2h 2 De De, Deh, Ded 2 d3 D3, D3d, D3h 2 T T, Th, Td 2 De De, Deh 1 I I, Ih 1 O O, Oh 1 Table 1: The classes of symmetry groups of ({va, ve}, 3)-sphere. Figure 1: The construction of GC2,1(Octahedron). degree 6. It works by breaking the triangles of M into smaller triangles and gluing the pieces together in order to get another triangulation. See an example on Figure 1 and [9] for more details. If M has nT triangles then GCk,l(M) has nT (k2 + kl +l2) triangles. Since k2 + kl + l2 = |k + lw|2 it makes sense to associate the Eisenstein integer z = k + lw to the pair k, l. The parameter symmetry z ^ zwr does not change the resulting map. All the cases of 1 parameter in Table 1 are described by the Goldberg-Coxeter construction of one plane graph. For example, if a ({v3 = 4,ve}, 3)-, ({v4 = 6,ve}, 3)-, ({v5 = 12, ve}, 3)-sphere is of symmetry (I,Ih), (O,Oh) or (T,Td) then it is of the form GCkil(Icosahedron), GCkj(Octahedron) or GCk l(Tetrahedron) [14, 9]. Additionally, ({v4 = 6,ve}, 3)-spheres of symmetry (De, Deh) are obtained as GCkj(Prism*6). 2.2 One case of 2 parameters: ({v5 = 12, v6}, 3)-spheres of symmetry D5 The 5-fold axis of such a sphere has to pass through a vertex of degree 5. There are 5 vertices of degree 5 around it; so, by 5-fold symmetry, 1 complex parameter is needed to Figure 2: The parameterization of fullerenes of symmetry D5, D5d or D5h in term of (z1, z2) e Z[w]2 and two parameter operations. describe them. Around those 5 vertices, there are 5 more vertices, so one more parameter is needed and then the last vertex is uniquely defined. If one applies the following operations to parameters • Operation 1: (z1, z2) ^ (z1, z1 + z2) • Operation 2: (z1, z2) ^ (z1 + J1z2, z1 — z2) • Operation 3: (z1,z2) ^ (z1 ,z2)wr then one obtains the same sphere as a result. The group generated by those operations is named monodromy group. The number of triangles of S is expressed as q(z1,z2) = 10{z1žT + ^glg*2 }. So, for a given pair (z1, z2) a sphere may not exist, for example, if q(z1, z2) < 0. 2.3 Other two parameters descriptions Of course what has been done for fullerene of class D5 applies just as well for fullerenes of class D6 and similar simple description are possible for the remaining 2 parameter cases of Table 1 (See Figure 3 for two such cases). For the ({v3 = 4, v6}, 3)-spheres a very explicit two parameter description is given in Figure 4. Geometrically this corresponds to the fact that any ({v3 = 4, v6}, 3)-sphere is obtained as the quotient of a ({v6}, 3)-torus by a group of order 2 leaving invariant exactly 4 vertices. Clearly, the monodromy group is PSL(2, Z) and the number of triangles is expressed as gzg(z1z2 — z[z2). This description was used in [21] to compute the eigenvalues of dual ({v3 = 4, v6}, 3)-spheres. Figure 3: Two parameters description of two classes of spheres with positive curvature represented on the plane. Figure 4: Description of ({v3 = 4, v6}, 3)-spheres. 3 Parameterization of maps on oriented surfaces 3.1 The general case The general Eberhard problem is for a given g > 0 and vector (v)3 0 for all face f of T and this defines the realizability space. The limit realizability space is the same space with Z[w] replaced by C. The limit realizability space defines a set S in the cone qT > 0. If one approach through a generic point of the boundary of S, which is not in the boundary of qT > 0 then one can rearrange the triangulation and get another triangulation T'. The vertex degree v in T may change but the degree of v remain the same. Similarly the cycles C are mapped to cycles C ' and I (C', a.') = I (C, a). Hence D and I are intrinsic to the class of triangulations obtained by rearrangements. Moreover, the parameter set (z') of (T', D, I, a') can be expressed linearly in term of the parameter set (zi) of (T, D, I, a). For two quadruples (T, D, I, a) and (T', D, I, a') of parameter set (zi), (z') we say that they are equivalent if there is a mapping from T to T' preserving D and I such that (zi) can be expressed linearly from (zi). Such an equivalence preserve edge length, triangle areas and can be extended to adjacent triangulations of T and correspondingly T'. Note that some non-trivial equivalence can have T = T'; this is the case for the 2 parameter description considered in Section 2 for which one triangulation suffices. The group of such transformation is the monodromy group and is a subgroup of GLm(Z[w]) leaving invariant the form qT. It may be that any two triples (T, D, I), (T', D, I) are related by a sequence of such transformation but we have no proof of it and we do not know a counter-example. 3.2 Thurston theory for maps of positive curvature Let us take one triple (v3, v4, v5) among the 19 triples of possible curvature. By Theorem 3.1 the number of complex parameters needed to describe ({v3, v4, v5, v6}, 3)-spheres is m = m({v3, v4, v5}, 0) = v3 + v4 + v5 - 2 The monodromy group is denoted by M({v3, v4, v5}, 3,0) and it preserves the form q, which is of signature (1, m - 1,0). This class of monodromy groups was defined and enumerated in [4,23,28] and in particular they are discrete groups. The 19 possible (v3, v4, v5) cases are part of the 94 cases determined there and the form q is the intersection form on H1 (S2 - V, L) with V a set of m + 2 points and L a line bundle on S2 - V. Figure 6: Three fullerenes of symmetry D5 or more. Figure 7: Two representations of a ({6}, 3)-torus. In [28] it is proved that if z G Z[w]m and q(z) > 0 then there exists f e M ({v3,v4,v5}, 3,0) such that f (z) is realizable as a ({v3, v4, v5, v6}, 3)-sphere. Thus Hm n Z[w]m up to the action of the monodromy group is a parameter space for the ({v3, v4, v5, v6}, 3)-spheres. As a consequence, the quotient Hm/(R>o X M3({v3,v4,v5}, 3,0)) is of finite covolume because the number of ({v3, v4, v5, v6}, 3)-spheres is finite for any fixed number of triangles. In [28] a characterization of the manifolds admitting a cocompact quotient is given. None of those corresponding to ({v3, v4, v5, v6}, 3)-spheres are compact. Each of the direction of non-compacity correspond to a partition of the vertices of non-zero curvature into two sets S for i = 1, 2 each having vq vertices of degree q and satisfying to 3v3 + 2v4 + v5 =6. Geometrically those are nanotubes, that is we have two caps Q with (v3, v4, v5 ) vertices separated by a number of rings of vertices of degree 6 (see Figure 6). 3.3 Extensions and other cases of parameter descriptions One extension that can be done relatively simply is to consider vertices of degree 1 or 2. For example, the ({v2 = 3, v6}, 3)-spheres are obtained by applying the Goldberg-Coxeter construction to the sphere reduced to a cycle of length 3 [18]. One example that does not fit exactly into the above scheme is the description of ({v6}, 3)-torus by two parameters. It is done by writing down a parallelogram on the plane and identifying the sides. Since all vertices are equivalent in the torus, we have to choose one vertices of degree 6 used in the construction. In Figure 7 we show two equivalent representation of a ({v6}, 3)-torus. One of the representation has one horizontal line in the fundamental domain; it is actually always possible to have such a representation and this is the basis of a 3 integral parameters construction [25]. Goldberg-Coxeter construction and complex parameterizations for ({vi,v2,v3}, 6)-spheres are derived in [6]. The method was to apply the truncation operation to each vertex in order to get a ({v2, v4, v6}, 3)-sphere for which existing theory could be applied. Yet another extension is for quadrangulations. The theory extends without difficulty, we are still dealing with triples (T, D, I) but the ring of Eisenstein integers is replaced by the ring of Gaussian integers. For example, for ({v3 = 8, v4}, 4)-sphere, the number of Gaussian integer parameters is 6 and the number for the classes (O, Oh), (D4, D4d, D4h), (Ds, D3d, Dsh), (D2, D2d, D2h), (C2, C2h, C2v), (Ci, Cs, Ci), respectively is 1, 2, 2, 3, 4, and 6. As a byproduct of this parameterization, we also get a method for parametrizing self-dual plane graphs with faces of size 3 or 4, see [10] for details. Finally note that all of the above can be specialized to get parameter description of families of maps having a specific symmetry group G provided that G contains only orientation preserving mappings. This is because the symmetry conditions are translated into linear equalities in the parameters. 4 Zigzag In an oriented map a zigzag is a circuit of edges such that two consecutive share a face and vertex but three do not share a face. 4.1 The Goldberg-Coxeter case For a triangulation M we define in [9] a permutation group Mov(M) and two elements L and R. If gcd(k, l) = 1 then the lengths of zigzags of GCkjl(M) is computed from the cycle structure of the element L ©kjl R of Mov(M). This element satisfies the defining relations L ©1j0 R = L, L ©0j1 R = R and If gcd(k, l) = m > 1 then every zigzag of GCk/mjl/m(M) corresponds to m zigzags of GCkji (M ) of length multiplied by m. The same method applies as well for 4-regular plane graphs and their central circuit, see [9] for an exhaustive description. 4.2 The case of ({v3 = 4, v6}, 3)-spheres All zigzags of ({v3 = 4,v6}, 3)-spheres are simple and the vector enumerating their lengths is of the form This was first established in [17] but there is another way to establish it: Any ({v3 = 4, v6}, 3)-sphere is obtained as a quotient of a ({v6}, 3)-torus by a group of order 2 formed L ©M R = L ©k-qi, i RLq if k - > 0, L ©k,i R = RqL ©k , i-qk R if I - qk > 0. (4s1)mi, (4s2)m2, (4ss)m3 with si,mi G N and simi n 4 Figure 8: The parameter description of a family of ({v4 = 6, v6}, 3)-sphere with simple zigzags, z1 = hu and z2 = h — 2k + ku and the case (h, k) = (4,1). by inversion. The four vertices of degree 3 come from the four invariant vertices of the torus. All zigzags of a ({v6}, 3)-torus are partitioned into three parallel classes that cover the vertex set. All the zigzags in a parallel class are of the same length and when passing to the quotient the parallel classes are preserved hence the above result is proved. The same argument applies for ({v2 = 4, v4}, 4)-spheres and their central circuits [8]. 4.3 Other classes For other classes of maps with more parameters the structure is more complicated and it seems very difficult to obtain simple description of the zigzags of fullerenes. However, for ({v4 = 6, v6}, 3)-spheres we have a simple conjecture for the ones with simple zigzags: Conjecture 4.1. All ({v4 = 6, v6}, 3)-spheres with only simple zigzags are: • GCk k (Octahedron) and • the family of graphs with parameters (m, k) with n = 4h(2h — 3k) triangles whose parameter description is given in Figure 8. The vector enumerating zigzag length is z = (6h — 6k)3h-3k, (6h)h-2k, (12h — 18k)k They have symmetry Oh if k = 0, D6h if h = 3k and D3d otherwise. References [1] G. Brinkmann, O. Delgado-Friedrichs, A. Dress and T. Harmuth, CaGe - a virtual environment for studying some special classes of large molecules, MATCH-Commun. Math. Co. 36 (1997), 233-237. [2] G. Brinkmann, T. Harmuth and O. Heidemeier, The construction of cubic and quartic planar maps with prescribed face degrees, Discrete Appl. Math. 128 (2003), 541-554. [3] M. Conder and P. Dobcsanyi, Determination of all regular maps of small genus, Combin. Theory Ser. B 81 (2001), 224-242. [4] P. Deligne and G.D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. Inst. Hautes Études Sci. 63 (1986), 5-89. [5] M. Deza and M. Dutour, Zigzag Structure of Simple Two-faced Polyhedra, Comb. Probab. Comput. 14 (2005), 31-57. [6] M. Deza and M. Dutour Sikiric, Zigzag and central circuit structure of ({1, 2, 3}, 6)-spheres, Taiwan. J. Math., to appear [7] M. Deza, M. Dutour Sikiric and P. Fowler, The symmetries of cubic polyhedral graphs with face size no larger than 6, MATCH-Commun. Math. Co. 61 (2009), 589-602. [8] M. Deza, M. Dutour and M.I. Shtogrin, 4-valent plane graphs with 2-, 3- and 4-gonal faces, "Advances in Algebra and Related Topics" (in memory of B.H. Neumann; Proceedings of ICM Satellite Conference on Algebra and Combinatorics, Hong Kong 2002), World Scientific Publ. Co. (2003), 73-97. [9] M. Dutour and M. Deza, Goldberg-Coxeter construction for 3- and 4-valent plane graphs, Electron. J. Combin. 11 (2004), R#20. [10] M. Dutour Sikiric and M. Deza, 4-regular and self-dual analogs of fullerenes, Mathematics and Topology of Fullerenes, pp. 103-116 ed. by O. Ori, A. Graovac and F. Cataldo, Carbon materials, Chemistry and Physics, vol 4, Springer Verlag, 2011. [11] M. Dutour Sikiric and P. Fowler, Cubic ramapolyhedra with face size no larger than 6, J. Math. Chem. 49 (2011), 843-858. [12] P. W. Fowler, J.E. Cremona and J. I. Steer, Systematics of bonding in non-icosahedral carbon clusters, Theory Chem. Acta 73 (1988), 1-26. [13] P. W. Fowler and D. E. Manolopoulos, An Atlas of Fullerenes, Clarendon Press, Oxford, 1995. [14] M. Goldberg, A class of multisymmetric polyhedra, Tohoku Math. J. 43 (1937), 104-108. [15] J. E. Graver, Encoding fullerenes and geodesic domes, SIAM J. Discrete Math. 17 (2004), 596614. [16] J. E. Graver, Catalog of all fullerenes with ten or more symmetries, Graphs and discovery, 167-188 DIMACS Series Discrete Mathematics Theoretical Computer Science, 69, American Mathematical Society, 2005. [17] B. Grunbaum and T. S. Motzkin, The number of hexagons and the simplicity of geodesics on certain polyhedra, Canadian J. Math. 15 (1963), 744-751. [18] B. Grunbaum and J. Zaks, The existence of certain planar maps, Discrete Math. 10 (1974), 93-115. [19] T. Harmuth, The construction of cubic maps on orientable surfaces, PhD thesis, Bielefeld, 2000. [20] S. Jendrol', On face vectors of trivalent maps, Math. Slovaca 36 (1986), 367-386. [21] P. E. John and H. Sachs, Spectra of toroidal graphs, Discrete Math. 309 (2009), 2663-2681. [22] H. W. Kroto, J. R. Heath, R. F. Curl, R. E. Smalley, C60: Buckminsterfullerene, Nature 318 (1985), 162-163. [23] G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Publ. Math. Inst. Hautes Etudes Sci. 63 (1986), 91-106. [24] G. D. Mostow, Braids, hypergeometric functions, and lattices, B. Am. Math. Soc. 16 (1987), 225-246. [25] S. Negami, Uniqueness and faithfulness of embedding of graphs into surfaces, Doctor thesis, Tokyo Institute of Technology, 1985. [26] I. Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. Math. 139 (1994), 553-580. [27] C. H. Sah, A generalized leapfrog for fullerene structures, Fullerene Sci. Techn. 2 (1994), 445458. [28] W. P. Thurston, Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift, Geom. Topol. Monogr. 1 (1998), 511-549. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 83-88 On the minimum rainbow subgraph number of a graph Ingo Schiermeyer Institut fur Diskrete Mathematik und Algebra, Technische Universität Bergakademie Freiberg, 09596 Freiberg, Germany Received 19 October 2011, accepted 9 April 2012, published online 1 June 2012 Abstract We consider the Minimum Rainbow Subgraph problem (MRS): Given a graph G whose edges are coloured with p colours. Find a subgraph F C G of minimum order and with p edges such that each colour occurs exactly once. This problem is NP-hard and APX-hard. For a given graph G and an edge colouring c with p colours we define the rainbow subgraph number rs(G, c) to be the order of a minimum rainbow subgraph of G with size p. In this paper we will show lower and upper bounds for the rainbow subgraph number of a graph. Keywords: Edge colouring, rainbow subgraph. Math. Subj. Class.: 05C15, 05C35 1 Introduction and motivation We use [2] for terminology and notation not defined here and consider finite and simple graphs only. Our research was motivated by the following problem from bioinformatics. The problem data consist in a set G of p genotypes g i, g2,..., gp corresponding to p individuals in a population. Each genotype g is a vector with entries in {0,1, 2}. Each position where a 2 appears is called ambiguous position. For a genotype g we have to determine a pair of haplotypes hP and hM (hP stands for the paternal haplotype and hM stands for the maternal haplotype), which are binary vectors such that g = hP © hM. Given two haplotypes h' and h'', their sum is defined as the vector g = h' © h" with g[i] = 0, if h'[i] = h''[i] = 0, g[i] = 1, if h'[i] = h''[i] = 1 and g[i] = 2, if h'[i] = h''[i]. We say that a set H of haplotypes resolves G if for every g e G there exist hi,h2 G H such that g = hi © h2. Given a set G of genotypes, the haplotyping problem consists E-mail address: Ingo.Schiermeyer@tu-freiberg.de (Ingo Schiermeyer) in finding a set H of haplotypes that resolves G. In the Pure Parsimony Haplotyping problem (PPH problem) we are interested in finding a set H of smallest possible cardinality. If each genotype has at most k ambiguous positions, then we denote this problem by PPH(k). The PPH problem has been studied in ([3],[4],[7],[9]). Matos Camacho et al. [8] have shown that the PPH(k) can be transformed to a graph problem, the Minimum Rainbow S ubgraph problem (MRS). Note that this edge-colouring need not be proper. Definition 1.1 (Rainbow subgraph). Let G be a graph with an edge-colouring. A subgraph H of G is called rainbow subgraph if H does not contain two edges of the same colour. Definition 1.2 (Minimum Rainbow Subgraph problem (MRS)). Given a graph G, whose edges are coloured with p colours, find a subgraph F C G of minimum order and with p edges such that each colour occurs exactly once. For a set G of p genotypes gi, g2,..., gp we will use p colours 1, 2,... ,p. For each haplotype we introduce a vertex. If two haplotypes hh and h" resolve a genotype gi (gi = hh © h" ), then the corresponding vertices will be joined by an edge which receives colour i. If a genotype is resolved by two identical haplotypes, then the corresponding vertex is joined by an edge which is called a loop. In this way we construct a graph G, whose edges are coloured with p colours. Note that this is a proper edge colouring (no vertex is incident with two edges of the same colour), since a haplotype h can be used at most once in a pair of haplotypes, which resolves a genotype g. Furthermore, every set H of haplotypes that resolves G corresponds to a rainbow subgraph F of G. It has been shown in [8] that a graph G containing loops can be transformed into a graph G' without loops. Hence in the following we may assume that all graphs have no loops. Matos Camacho et al. [8] proved the MRS problem to be NP-hard and APX-hard. In [5] it has been shown that the MRS problem remains NP-hard and APX-hard even for graphs with maximum degree 2. Remark: If we do not consider edge colourings, the analogous problem is known as the (t, f (t)) dense subgraph problem ((t, f (t))-DSP), which asks whether there is a t-vertex subgraph of a given graph G which has at least f (t) edges. When f (t) = Q, (t, f (t))-DSP is equivalent to the well-known t-clique problem (cf. [1]). 2 Lower bounds for the rainbow subgraph number Definition 2.1. Let G be a graph and c be its edge colouring with p colours. The rainbow subgraph number of G (with respect to the colouring c) is defined as the order of its minimum rainbow subgraph of size p, and denoted by rs(G, c) (or rs(G), when the colouring c is clear from the context). Improved lower bounds for the rainbow subgraph number rs(G) will be of major importance for the design of approximation algorithms with better approximation ratios for the MRS problem (cf. [8, 5]). So far nothing better than the trivial lower bound rs(G) > —G) is known. We can improve this lower bound by counting the number of distinct colours among all edges incident to a vertex. Definition 2.2. Given an edge colouring of a graph G with colours 1, 2,... ,p, we define c(e) = i, if the edge e has colour i for 1 < i < p. Let cd(v) (colour degree) denote the number of distinct colours among all edges incident to the vertex v and let cd(i) = ma x{cd(v) | v G V (G) has an incident edge with colour i} be the maximum colour degree for every colour i, 1 < i < p. Using the maximum colour degrees for all colours we can show the following improved lower bound. Proposition 2.3. Let G be a graph, whose edges are coloured with p colours. Then p >(G - E 2 2p > i=i cd(i) - A (G) Proof. Let F be a minimum rainbow subgraph of order k = rs(G). Then rs(G) = k = £ m= £ + ^T - t — > 2P dF(v) dF(u) dF(w) cd(i) A(G) veV (F ) FW e=uw,e£E(F ) FV 7 FV 7 i=1 W V ' □ The following example shows that this bound is sharp and improves the lower bound of AAG significantly. Example 2.4. For p - 4 and A - 2 let G = Ki,a + Cp-i (where G + H denotes the disjoint union of two graphs G and H). All edges of the cycle Cp-1 are coloured distinctly, say with colours 1, 2,... ,p - 1, and all edges of K1 A are coloured with colour p. Then rs(G) = p + 1= p - 1 + 2 = EP=i 4) > AG). ' We can further improve this lower bound by counting the number of distinct colours among all edges incident to the endvertices of an edge. For this purpose we define q(i) = min{cdu) + cdww)1 uw G E(G) and c(uw) = i}. Proposition 2.5. Let G be a graph, whose edges are coloured with p colours. Then ^ ^ 2 2p rs(G) - t q(i -1 cd« - am. Proof. Let F be a minimum rainbow subgraph of order k = rs(G). For every colour i, 1 < i < p, let ruiwi be an edge such that ed(U. ) + = q(i). If 'uw G E(F) is an edge with c(uw) = i, then + J > } -, + = q(i) > 2 ■-r ,, \ —^ > . v ' 1 ea(u) ea(w) — ea(ui) ea(wi) ' — max{ed(ui),ed(wi)} — ed(i) Therefore, dF(v) 1 1 rs{G)=k = E dm= e dF ( v) dF ( u) dF ( w) veV (F ) F W e=uw,e£E(F ) ' F V ' > ^ A 2 2p gq(i) - g ^ - AM. □ The following example shows that this bound is sharp and improves the previous two lower bounds significantly. Example 2.6. Let G = KijP for some p > 2. Let the edges of G be coloured with p colours. Then cd(i) = p and q(i) = 1 + P for 1 < i < p. Thus rs(G) = p + 1 = p ■ (1 + p ) = EPU q(i) > 2 = p ■ P = EpU = sdb. 3 Upper bounds for the rainbow subgraph number First observe that the trivial upper bound rs(G) < 2p is achieved if the rainbow subgraph F is a matching. This upper bound has been improved towards rs(G) < 2p +1 - A(G) by Koch [6] for properly edge-coloured graphs and this bound is sharp. For instance, let G = Ki,a + (p - A)K2, where p > A, and all edges of G are coloured distinctly. Then rs(G) = 2p + 1 - A(G). Similar to Brooks' Theorem (cf. [2]) we can characterize all graphs achieving this bound. Theorem 3.1. Let G be a graph with maximum degree A > 2, whose edges are properly coloured with p colours. If rs(G) = 2p +1 — A(G), then G has the following properties: 1. G contains a star K1jA with center vertex v0 and leaves v1,..., vA and G[N(v0)] is edgeless. Let c(v0vj ) = i for 1 < i < A and H0 = G[N [v0]]. 2. If p > A, then let Hi be the subgraph spanned by the edges with colour i for A + 1 < i < p. The subgraphs Ha+1, Ha+2, ... ,Hp are pairwise vertex-disjoint and V(H0) n V(Hi) = 0 for A + 1 < i < p. 3. E (Hi, Hj ) = 0 for A + 1 < i < j < p (where E(Hi, H j ) is the set of all edges having one vertex in V(Hi) and the other vertex in V(Hj)). 4. E (vi, Hj ) = 0 for 1 < i < A and A+1 < j < p (where E(vi, H j ) is the set of all edges incident with vi and a vertex in V(Hj )). 5. If uv e E (Hi) for some A + 1 < i < p, then N (u) n N (v) = 0. 6. N (vi) n N (vj ) = 0 for vi e V (Hi), v j e V (Hj ), A + 1 < i < j < p. Proof. 1. Suppose there is an edge vivj- for some 1 < i < j < A. If c(vivj-) = k for some k with 1 < k < A, k = i, j, thenrs(G) < (A+1)-1 + (2p-2A) = 2p-A < 2p +1 — A, a contradiction. If c(vivj- ) = k for some k with A+1 < k < p, then rs(G) < (A + 1) + (2p - 2A - 2) = 2p - A - 1 < 2p +1 - A, a contradiction as well. 2. Suppose there are integers i, j with A + 1 < i < j < p and two adjacent edges e, f with c(e) = i, c(/) = j. Then rs(G) < (A + 1) + (2p - 2A - 1) = 2p -A < 2p +1 - A, a contradiction. Suppose there are integers i, j with 1 < i < A, A+1 < j < p and two adjacent edges e, / with c(e) = i, c(/) = j. Then rs(G) < (A + 1) + (2p - 2A - 1) = 2p - A < 2p +1 - A, a contradiction as well. 3. Suppose there is an edge vivj- with vi e V(Hi), vj e V(Hj), A + 1 < i < j < p. Then c(vivj) = k for some 1 < k < A. Hence rs(G) < (a +1) -1 + (2p - 2A) = 2p - A < 2p +1 - A, a contradiction. 4. Suppose there is an edge vivj- for two vertices vi e V(H0) and vj e V(Hj ), A+1 < j < p. Then rs(G) < (A+1) + (2p-2A-1) = 2p-A < 2p+1-A, a contradiction. 5. Suppose there is an edge uv e E (Hj) for some A +1 < i < p with N (u) n N (v) = 0. By 3. and 4. we conclude that N (u) n N (v) n V (H0) = 0. Furthermore, for a vertex w e N (u) n N (v), we have c(uw) = j, c(vw) = k for some 1 < j < k < A. Then rs(G) < (A+l) - 2 + (2p - 2A+1) = 2p - A < 2p +1 - A, a contradiction. 6. Suppose N (vi) n N (vj ) = 0 for two vertices vi e V (Hi ),vj e V (Hj ), A+1 < i < j < p. By 3. and4. we conclude that N (vi)nN (vj )nV (H0) = 0. Furthermore, for a vertex w e N(vi)nN(vj ), we have c(uw) = k, c(vw) = l for some 1 < k < l < A. Then rs(G) < (A+1) - 2 + (2p - 2A+1) = 2p - A < 2p +1 - A, a contradiction. □ Another upper bound for the rainbow subgraph number follows from an approach presented in [8]. Observe that two adjacent edges of different colours together have three vertices, whereas two edges of different colours in a matching have four vertices. Based on this obervation the following algorithm has been proposed in [8]. Algorithm Input: A graph G of order n whose edges are coloured with p colours 1. Construct a graph G' with V(G') = jvi, v2,..., vp} (vi corresponds to colour i) and vivj e E (G') if there exist two adjacent edges e, f e E(G) with c(e) = i and c(f ) = j (c(x) denotes the colour of the edge x). 2. Now compute a maximum matching M of order ß(G') in G'. This can be done in polynomial time. 3. Next construct a graph H with V (H ) C V (G) as follows: For each matching edge of M choose two adjacent edges in G with these two colours. For each vertex of V (G') not in M choose an edge in G with this colour. In this way we obtain a rainbow subgraph H C G with |E(H)| = p. Correctness of the algorithm: Edges of the matching correspond to pairs of adjacent edges in the original graph. Colours that are left out by this procedure are added greedily at the end. Claim 3.2. |V(H)| < 2p - ß(G') Proof. For each matching edge of G' three vertices appear in H. Hence |V(H)| < 3ß(G') + 2(p - 2ß(G')) = 2p - ß(G') □ Corollary 3.3. rs(G) < 2p - ß(G'). Acknowledgement We thank the referees for some valuable comments. References [1] Y. Asahiro, R. Hassin and K. Iwama, Complexity of finding dense subgraphs, Discrete Appl. Math. 121 (2002), 15-26. [2] J. A. Bondy and U.S.R. Murty, Graph Theory, Springer, London, 2008. [3] D. Catanzaro and M. Labbe, The pure parsimony haplotyping problem: overview and computational advances, International Transactions in Operational Research 16 (2009), 561-584. [4] D. Gusfield, Haplotype inference by pure Parsimony, in: Proceedings ot the 14th annual conference on Combinatorial pattern matching (CPM'03), Springer, Berlin, Heidelberg, 2003, 144155. [5] J. KatreniC and I. Schiermeyer, Improved approximation bounds for the minimum rainbow subgraph problem, Inf. Process. Lett. 111 (2011), 110-114. [6] M. Koch, Das Population-Haplotyping-Problem: Graphentheoretische Ansätze, Diploma thesis, TU Bergakademie Freiberg, 2008. [7] G. Lancia, M. C. Pinotti and R. Rizzi, Haplotyping Populations by Pure Parsimony: Complexity of Exact and Approximation Algorithms, INFORMS Journal on Computing 16 (2004), 348-359. [8] S. Matos Camacho, I. Schiermeyer and Z. Tuza. Approximation algorithms for the minimum rainbow subgraph problem, Discrete Math. 310 (2010), 2666-2670. [9] L. S. Wang, Hapolotype inference by maximum parsimony, Bioinformatics 19 (2003), 17731780. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 89-97 A note on domination and independence-domination numbers of graphs* Martin MilaniC University of Primorska, UP IAM, Muzejski trg 2, SI6000 Koper, Slovenia, and University of Primorska, UP FAMNIT, Glagoljaška 8, SI6000 Koper, Slovenia Received 30 November 2011, accepted 25 April 2012, published online 1 June 2012 Abstract Vizing's conjecture is true for graphs G satisfying y®(G) = 7(G), where 7(G) is the domination number of a graph G and 7®(G) is the independence-domination number of G, that is, the maximum, over all independent sets I in G, of the minimum number of vertices needed to dominate I. The equality 7® (G) = 7(G) is known to hold for all chordal graphs and for chordless cycles of length 0 (mod 3). We prove some results related to graphs for which the above equality holds. More specifically, we show that the problems of determining whether y®(G) = 7(G) = 2 and of verifying whether y®(G) > 2 are NP-complete, even if G is weakly chordal. We also initiate the study of the equality 7® = 7 in the context of hereditary graph classes and exhibit two infinite families of graphs for which Y ® < Y. Keywords: Vizing's conjecture, domination number, independence-domination number, weakly chordal graph, NP-completeness, hereditary graph class, IDD-perfect graph. Math. Subj. Class.: 05C69, 68Q17 1 Introduction The closed neighborhood NG [v] of a vertex in a (finite, simple, undirected) graph G is the set consisting of v itself and its neighbors in the graph. A set A of vertices is said to dominate a set B if B C U{NG [a] : a G A}. The minimum size of a set of vertices dominating a set A is denoted by yg (A). A dominating set in a graph G is a set D of vertices that dominates V(G). We write y(G) for yg(V(G)). The concept of domination in graphs has been extensively studied, both in structural and algorithmic graph theory, because of its numerous applications to a variety of areas. Domination naturally arises in facility location *This work is supported in part by "Agencija za raziskovalno dejavnost Republike Slovenije", research program P1-0285 and research projects J1-4010, J1-4021 and N1-0011. E-mail address: martin.milanic@upr.si (Martin Milanic) problems, in problems involving finding sets of representatives, in monitoring communication or electrical networks, and in land surveying. The two books [14,15] discuss the main results and applications of domination in graphs. Many variants of the basic concepts of domination have appeared in the literature. Again, we refer to [14,15] for a survey of the area, and to [4,10,11,13,16,18,19,21,22] for some recent papers on domination and variants thereof. The Cartesian product of two graphs G and H is the graph GOH with vertex set V (G) X V (H ) and edge set {(u, x)(v, y) : (u, x), (v, y) e V (G) x V (H ), u = v and xy e E(H), or x = y and uv e E (G)}. In 1968 Vizing made the following conjecture, according to Bresar et al. [8] "arguably the main open problem in the area of domination theory": Conjecture 1. For every two graphs G and H, it holds that 7(GOH) > 7(G)y(H). The conjecture is still open and was verified for several specific classes of graphs; see, e.g., [8]. An independent set in a graph is a set of pairwise non-adjacent vertices. The independence-domination number 7i(G) is the maximum of 7G(I) over all independent sets I in G. The independence-domination number has arisen in the context of matching theory, see, e.g., [2,20], and was first introduced in the context of domination by Aharoni and Szabo in 2009 [3]. Obviously, 7®(G) < 7(G), and in general the gap between the two may be large [3]. However, equality holds for: • cycles of length 0 (mod 3), and more generally, for graphs that have a set of 7(G) vertices with pairwise disjoint closed neighborhoods [17]; • chordal graphs, as proved by Aharoni, Berger and Ziv [1] in a result on width and matching width of families of trees. Recall that a graph G is called chordal if it does not contain any induced cycle of length at least 4, and weakly chordal if it does not contain any induced cycles of length at least 5 or their complements. Theorem 2 ( [1]). For every chordal graph G, it holds that 7®(G) = 7(G). The independence-domination number is related to Vizing's conjecture via the following result proved by Aharoni and Szabo [3]: Theorem 3 ( [3]). For every two graphs G and H, it holds that 7(GOH) > 7i(G)y(H). In particular, Vizing's conjecture is true for chordal graphs. More generally, if G is a graph with 7®(G) = 7(G) then y(GOH) > 7(G)y(H) for every graph H. In a recent survey paper on Vizing's conjecture [8], Bresar et al. asked what other classes of graphs can be found for which 7®(G) = 7(G) for every G in the class. In this note, we prove some results related to graphs for which the independence-domination number coincides with the domination number. First, using a relationship between the independence-domination number and the notion of a dominating clique, we prove that determining whether 7®(G) = 7(G) is NP-hard. More specifically, we show that it is NP-complete to determine whether 7®(G) > 2, as well as to determine whether 7®(G) = 7(G) = 2. These results, obtained in Section 2, remain valid for weakly chordal graphs. In Section 3, we turn our attention to graphs in which the equality y® = 7 holds in the hereditary sense. We show that this class, which properly contains the class of chordal graphs, is properly contained in the class of graphs in which all induced cycles are of length 0 (mod 3). We do this by constructing an infinite family of graphs in which all induced cycles are of length 0 (mod 3) but where the independence-domination number is strictly smaller than the domination number. In conclusion, we propose three related problems. 2 The complexity of computing 71 and testing 71 = 7 In this section, we study some computational complexity aspects of computing the indepedence-domination number and comparing it to the domination number. We first recall some notions needed in our proofs. For a graph G = (V, E), we denote by G its complement, that is, the graph with the same vertex set as G, in which two vertices are adjacent if and only if they are not adjacent in G. A clique in a graph is a subset of pairwise adjacent vertices. A dominating set that is also a clique is called a dominating clique. We assume familiarity with basic notions of computational complexity (see, e.g., [12]). Theorem 4. Given a weakly chordal graph G, it is NP-complete to determine whether Y®(G) > 2. Proof. To show membership in NP, observe that a short certificate for the fact that y® (G) > 2 is any independent set I such that for every vertex v G V (G), it holds that I ^ NG [v]. To show hardness, we make a reduction from the problem of determining whether a given weakly chordal graph contains a dominating clique. This is an NP-complete problem, see, e.g., [6]. Clearly, the problem remains NP-complete if we assume that the input graph G does not have a dominating vertex. Suppose that we are given a weakly chordal graph G without dominating vertices. We compute its complementary graph H = G. Since H is also weakly chordal, the theorem follows immediately from the claim below. Claim: G has a dominating clique if and only if y®(H) > 2. For the forward implication, suppose that G has a dominating clique K. We will show that Y ®(H) > 2 by showing that yh (K) > 2. Suppose for a contradiction that yh (K) = 1. Then, there exists a vertex v G V (H ) = V (G) such that K C NH [v]. In particular, v must belong to K, since otherwise in G, vertex v would not have any neighbors in K, contrary to the assumption that K is dominating in G. Since K is independent in H, that facts that v G K and K C NK [v] imply that K = {v}, that is, v is a dominating vertex in G, which is impossible since we assumed that G has no dominating vertices. Hence, it holds that YH (K) > 2 and consequently y®(H) > 2. For the converse implication, suppose that y®(H) > 2, and let I be an independent set in H such that yh(I) > 2. Clearly, I is a clique in G, and, in fact, a dominating clique: If this were not the case, then there would exist a vertex v g V (G) \ I such that in G, vertex v is not adjacent to any vertex from I. Equivalently, for every u G I, uv G E (H ). But then {v} would dominate I in H, contrary to the assumption that yh (I) > 2. □ Corollary 5. Given a (weakly chordal) graph G and an integer k, it is NP-hard to determine whether y® (G) > k. Corollary 6. Given a (weakly chordal) graph G and an integer k, it is NP-hard to determine whether y® (G) < k. How difficult it is to determine whether the values of y® and y coincide? Since 7® (G) < Y(G) holds for every graph G, in order to show that Y®(G) = y(g) = k, (2.1) it suffices to argue that y®(G) > k and y(G) < k. Clearly, for k = 1, whether (2.1) holds can be determined in polynomial time: a necessary and sufficient condition for y® (G) = Y(G) = 1 is that G has a dominating vertex. We now show that already for k = 2, the problem becomes NP-complete, even for weakly chordal graphs. The proof will also imply intractability of the problem of verifying whether y® = Y. Theorem 7. Given a weakly chordal graph G, it is NP-complete to determine whether Y ®(G) = Y (G) = 2. Proof. Membership in NP follows from the fact that a short certificate for y® ( G) = y ( G) = 2 is given by a pair (I, D) where I is an independent set not dominated by any vertex (proving y®(G) > 2) and D is a dominating set of size two (proving y(G) < 2). To show hardness, we make a reduction from 3-SAT [12]. The reduction is an adaptation of the reduction by Brandstadt and Kratsch [6] used to prove that the dominating clique problem is NP-complete for weakly chordal graphs. Suppose that we are given an instance to 3-SAT, that is, a Boolean formula ^ over variables xi,..., xn, consisting of m clauses of length 3, say C® = x^'1 V x^2 V i"'3 for i = 1,..., m, where a^ G {0,1}, with the usual interpretation that i1 = x® and x0 = x®. Without loss of generality, we may assume the following properties of the formula: Property 1: No clause contains both a literal and its negation. (This is because clauses containing both a literal and its negation can be discarded as they will always be satisfied.) Property 2: There exist two clauses, say C1 and C2, that have no literals in common. (If the given formula ^ does not have this property, we simply add to it a new clause consisting of three new variables. If necessary, we relabel the clauses.) Consider the graph H defined as follows: v(h) = {xi,xr,... ,x„,xn} u {Ci,... ,Cm}, E(H ) = {x«1 xa2 | 1 < i,j < n, i = j, ai,a2 G {0,1}}U {xf C 11 < i < n, 1 < j < m, a G {0,1}, xa is a literal in Cj} . We complete the reduction by computing the complementary graph G = H. Using Property 1, it is easy to verify that neither H nor G contain an induced cycle of length at least 5, that is, G is weakly chordal. Moreover, the following properties are equivalent: (i) ^ is satisfiable. ( ii) H has a dominating clique. (iii) Y®(G) = y(G) = 2. (iv) Y®(G) = y(G). The equivalence between (i) and (ii) has been established in [6]. (ii) implies (iii): Suppose that H has a dominating clique. Since H has no dominating vertex, similar arguments as in the proof of Theorem 4 allow us to conclude that y® (G) > 2. Furthermore, by Property 1 and by construction of H, vertices C1 and C2 have no common neighbors in H. This implies that {C1, C2} is a dominating set in G. Therefore y(G) < 2, and the conclusion follows since 2 < 7®(G) < 7(G) < 2. Trivially, (iii) implies (iv). (iv) implies (ii): Suppose that 7®(G) = 7(G). Since H has no isolated vertices, G has no dominating vertices. Therefore 7®(G) = 7(G) > 2, and it can be shown, similarly as in the proof of Theorem 4, that H has a dominating clique. This completes the proof. □ Theorem 8. Given a weakly chordal graph G, it is NP-hard to determine whether 7® (G) = Y (G). Proof. Perform the same reduction as in the proof of Theorem 7 and use the fact that the formula is satisfiable if and only if y ®(G) = y (G). □ 3 A hereditary view on 71 = 7 In this section, we initiate the study of the equality between the domination and independence-domination number of graphs in the context of hereditary graph classes. A graph class is said to be hereditary if it is closed under vertex deletions. The family of hereditary graph classes is of particular interest, first of all, since many natural graph properties are hereditary, and second, since hereditary (and only hereditary) classes admit a uniform description in terms of forbidden induced subgraphs. For a set F of graphs, we say that a graph G is F-free if it does not contain an induced subgraph isomorphic to a member of F. The set of all F-free graphs will be denoted by Free(F). Notice that for two sets F1 and F2 of graphs, it holds that Free(F1 U F2) = Free(F1) n Free(F2). Given a hereditary class G, denote by F the set of all graphs G with the property that G G G but H G G for every proper induced subgraph H of G. The set F is said to be the set of (minimal) forbidden induced subgraphs for G, and G is precisely the class of F-free graphs. The set F can be either finite or infinite, and many interesting classes of graphs can be characterized as being F-free for some family F. Such characterizations can be useful for establishing inclusion relations among hereditary graph classes, and were obtained for numerous graph classes (see, e.g. [7]). The most famous such class is probably the class of perfect graphs, for which the forbidden induced subgraph characterization is given by the Strong Perfect Graph Theorem conjectured by Berge in 1961 [5] and proved by Chudnovsky, Robertson, Seymour and Thomas in 2006 [9]. Since Vizing's conjecture holds for graphs G such that y®(G) = y(G), it would be interesting to determine the largest hereditary class of graphs with this property. Moreover, since recognizing graphs with y® = Y is NP-hard, it would also be interesting to determine whether graphs in which the property y® = Y holds in the hereditary sense can be recognized efficiently. With this motivation in mind, we introduce the class of independence-domination-domination-perfect graphs, or shortly, IDD-perfect graphs, that is, graphs for which the above equality holds in the hereditary sense: IDD-perfect graphs = {G : y®(H) = y(H) for every induced subgraph H of G} . We now provide some partial results towards a characterization of IDD-perfect graphs. By Theorem 2, we can immediately relate the class of IDD-perfect graphs to a well studied hereditary subclass of perfect graphs, the class of chordal graphs: Theorem 9. Chordal graphs C IDD-perfect graphs. Proof. Since every induced subgraph of a chordal graph is chordal, Theorem 2 implies that the class of iDD-perfect graphs contains the class of chordal graphs. This inclusion is proper since chordless cycles of length congruent to 0 (mod 3) are IDD-perfect [17] (but not chordal). □ In the rest of this section, we bound the class of IDD-perfect graphs from above, by exhibiting two infinite families of graphs that do not belong to class of IDD-perfect graphs: the chordless cycles of length not congruent to 0 (mod 3) and another graph family, which we describe now. For positive integers k1,k2,k3 > 1, let Fkl,k2,k3 denote the graph obtained from the disjoint union of three cycles C1, C2 and C3 where |V(Cj)| = 3kj as follows: denoting by (vj,..., vjk ) a cyclic order of vertices of Cj, we identify vertex v2 with vertex vfki , vertex vf with vertex v|k , and vertex vj with vertex v|k . See Fig. 1 for an example. Figure 1: The graph F2,2,2 Theorem 10. IDD-perfect graphs C Free ( U {Cfk+i,Cfk+2} U |J {Fk^^M . ^k>l ki,k2,k3 >1 ' Proof. First, we establish the inclusion IDD-perfect graphs C Free( |Jk> 1{C3k+1, C3k+2|). To this end, we show that for every chordless cycle C of order n = 3k +1 or n = 3k + 2 (where k is a positive integer), it holds that Yl(C) = k and y(C) = k + 1. Let (v1;..., vn) be a cyclic order of the vertices of such a cycle C. Observe that for every set S C V (C ) with |S | < k, it holds that |Uv£s Nc(v)| < ^ |Nc[v]| = 3|S| k + 1. On the other hand, the set {vfj-2 : 1 < j < k +1} is dominating, proving y(C) = k + 1. Suppose now that I is an independent set in C. We may assume w.l.og. that vi G I. In case that n = 3k + 2, we may also assume that vn G I. In either case, the set {v3j : 1 < j < k} is a set of size k dominating I. This shows that y®(C) < k. Conversely, since the set I = {v3j- : 1 < j < k} is a set of k vertices with pairwise disjoint closed neighborhoods, we have y®(C) > yc(I) = |I| = k. Thus k = y®(C) < y(C) = k + 1 and hence no IDD-perfect graph can contain C as an induced subgraph. It remains to show that IDD-perfect graphs C Free( |Jk fc ^i>1{Fk1,k2,k3 }) . Equiv-alently, we must show that for every three integers k1,k2,k3 > 1, it holds that Y®(FklMM) < Y(Fkl,k2,k3). We will show this in two steps, by computing the exact values of Yi'(FkiMM) and y(FkiMM). Let F = Fkl,k2,k3 for some k1; k2, k3 > 1. First, we show that y(F) = k1+k2+k3 — 1. Consider the set D = {v1j-2 : 1 < j < k1} U {v3j—1 : 1 < j < k2} U {v3j : 1 < j < k3 — 1} . Then, D is a dominating set of size k1 + k2 + k3 — 1, showing that y(F) < k1 + k2 + k3 — 1. Now, we show that y(F) > k1 + k2 + k3 — 1. Suppose for a contradiction that D is a dominating set in F with |D| < k1 + k2 + k3 — 2. Clearly, for every p G {1,2,3}, we have that |D n V(Cp)| > kp. Moreover, D must contain at least kp — 1 vertices from Cp other than vf and vfkp since otherwise not all vertices in the set {v^^ : 2 < j < kp} can be dominated by D. This implies that |D n { v 1, v 2, v 3}| — 1. We may assume without loss of generality that D n {v 1, v2, v3} = {v 1}. But this implies that |D n V (C2 )| = k2 — 1, a contradiction. Hence y(F) = k 1 + k2 + k3 — 1. In the rest of the proof, we show that y®(F) = k1 + k2 + k3 — 2 . Consider the set I = H- : 1 < j < k1} U {v2j-2 : 1 < j < k2} U {v3j : 1 < j < k3 — 1} . This is is a set of k1 + k2 + k3 — 2 vertices with pairwise disjoint closed neighborhoods. Therefore y® (F) > |I| = k1 + k2 + k3 — 2 . To see that y® (F) < k1 + k2 + k3 — 2, we will verify that yf (I) < k1 + k2 + k3 — 2 for every independent set I in F. Up to symmetry, it is sufficient to consider the following two cases: • Case 1: v1 G I. In this case, the set D = {v1j-2 : 2 < j < k1} U {v3j : 1 < j < k2} U {v3j_2 : 2 < j < k3} is a set of size k1 + k2 + k3 — 2 dominating I. • Case2: {v1,v31ki-1,v2,vlk2-1,v3,vfk3-1} C I. In this case, the set D = {v 1, v 2, v 3} U {v1j- 1 : 2 < j < k1 — 1}U{v2j-1 : 2 < j < k2 — 1}U {v3j- 1 : 2 < j < k3 — 1} is a set of size k1 + k2 + k3 — 3 dominating I. This shows that k1 + k2 + k3 — 2 = y®(F) < y(F) = k1 + k2 + k3 — 1 and hence no IDD-perfect graph in can contain F = Fkl,k2,k3 as an induced subgraph. This completes the proof. □ Remark. Theorem 10 shows that the class of iDD-perfect graphs is not comparable with the class of perfect graphs. On the one hand, the 9-cycle is an IDD-perfect graph that is not perfect. On the other hand, the 4-cycle is a (bipartite, hence) perfect graph that is not IDD-perfect. 4 Conclusion We conclude this note with three problems related to results from Section 3. Problem 1. Determine whether every graph of the form Fkl ,k2,k3 is a minimal forbidden induced subgraph for the class of IDD-perfect graphs. Problem 2. Determine the set of minimal forbidden induced subgraphs for the class of IDD-perfect graphs. Problem 3. Determine the computational complexity of recognizing IDD-perfect graphs. Acknowledgements The author would like to thank Douglas Rall for stimulating discussions, and the three anonymous referees for comments that helped to improve the presentation of the paper. In addition, one referee suggested a significant simplification of a part of the proof of Theorem 10. References [1] R. Aharoni, E. Berger and R. Ziv, A tree version of Konig's theorem, Combinatorial 22 (2002), 335-343. [2] R. Aharoni and P. Haxell, Hall's theorem for hypergraphs, J. Graph Theory 35 (2000), 83-88. [3] R. Aharoni and T. Szabó, Vizing's conjecture for chordal graphs, Discrete Math. 309 (2009), 1766-1768. [4] G. Bacsó, Complete description of forbidden subgraphs in the structural domination problem, Discrete Math 309 (2009), 2466-2472. [5] C. Berge, Farbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. [6] A. Brandstadt and D. Kratsch, On domination problems for permutation and other graphs, Theoret. Comp. Sci. 54 (1987), 181-198. [7] A. Brandstadt, V. B. Le and J. Spinrad, Graph classes: a survey. SIAM Monographs on Discrete Mathematics and Applications. SIAM, Philadelphia, PA, 1999. [8] B. Bresar, P. Dorbec, W. Goddard, B. L. Hartnell, M. A. Henning, S. Klavzar, D. F. Rall, Vizing's conjecture: a survey and recent results, J. Graph Theory 69 (2012), 46-76. [9] M. Chudnovsky, N. Robertson, P. Seymour and R. Thomas, The strong perfect graph theorem, Ann. of Math. 164 (2006), 51-229. [10] F. Dahme, D. Rautenbach and L. Volkmann, Some remarks on a-domination, Discuss. Math. Graph Theory 24 (2004), 423-430. [11] J. E. Dunbar, D. G. Hoffman, R. C. Laskar and L. R. Markus, a-Domination, Discrete Math. 211 (2000), 11-26. [12] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, 1979. [13] F. Harary and T. W. Haynes, Double domination in graphs, Ars Combin. 55 (2000), 201-213. [14] T. W. Haynes, S. Hedetniemi and P. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, 1998. [15] T. W. Haynes, S. Hedetniemi and P. Slater (Eds.), Domination in Graphs: Advanced Topics, Marcel Dekker, 1998. [16] M. A. Henning and A. P. Kazemi, fc-tuple total domination in graphs, Discrete Appl. Math. 158 (2010), 1006-1011. [17] M. S. Jacobson and L. F. Kinch, On the domination of the products of graphs. II. Trees, J. Graph Theory 10 (1986), 97-106. [18] R. Klasing and C. Laforest, Hardness results and approximation algorithms of fc-tuple domination in graphs, Inform. Process. Lett. 89 (2004), 75-83. [19] H. Liu, M.J. Pelsmajer, Dominating sets in triangulations on surfaces, Ars Math. Contemp. 4 (2011), 177-204. [20] R. Meshulam, The clique complex and hypergraph matching, Combinatorica 21 (2001), 89-94. [21] O. Schaudt, On the existence of total dominating subgraphs with a prescribed additive hereditary property, Discrete Math. 311 (2011), 2095-2101. [22] Zs. Tuza, Hereditary domination in graphs: Characterization with forbidden induced subgraphs, SIAM J. Discrete Math. 22 (2008), 849-853. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 99-115 On D. G. Higman's note on regular 3-graphs Daniel Kalmanovich Department of Mathematics Ben-Gurion University of the Negev 84105 Beer Sheva, Israel Received 17 October 2011, accepted 25 April 2012, published online 4 June 2012 Abstract We introduce the notion of a t-graph and prove that regular 3-graphs are equivalent to cyclic antipodal 3-fold covers of a complete graph. This generalizes the equivalence of regular two-graphs and Taylor graphs. As a consequence, an equivalence between cyclic antipodal distance regular graphs of diameter 3 and certain rank 6 commutative association schemes is proved. New examples of regular 3-graphs are presented. Keywords: Antipodal graph, association scheme, distance regular graph of diameter 3, Godsil-Hensel matrix, group ring, Taylor graph, two-graph. Math. Subj. Class.: 05E30, 05B20, 05E18 1 Introduction This paper is mainly a clarification of [6] — a short draft written by Donald Higman in 1994, entitled "A note on regular 3-graphs". The considered generalization of two-graphs was introduced by D. G. Higman in [5]. As in the famous correspondence between two-graphs and switching classes of simple graphs, t-graphs are interpreted as equivalence classes of an appropriate switching relation defined on weights, which play the role of simple graphs. In his note Higman uses certain association schemes to characterize regular 3-graphs and to obtain feasibility conditions for their parameters. Specifically, he provides a graph theoretic interpretation of a weight and from the resulted graph he constructs a rank 4 symmetric association scheme and a rank 6 fission of it. Furthermore, he proves that rank 6 schemes with parameters as in his construction are equivalent to regular 3-graphs. During our redetermination of the structure constants of the rank 6 scheme an error in [6] was detected, this miscalculation led Higman to a false restriction on the parameters E-mail address: dannykal@bgu.ac.il (Daniel Kalmanovich) of regular 3-graphs. Our first contribution is the correction of this mistake (see Subsection 5.2). The second contribution is a proof (see Section 4) that in the case of regular 3-graphs, the graph defined by a weight in its switching class is a distance regular cover of the complete graph. Moreover, it is a cyclic antipodal distance regular (ADRG) 3-fold cover of the complete graph in the sense of Godsil and Hensel in [2]. This provides a further restriction on the parameters of regular 3-graphs. Altogether, in Section 4 and in Section 5 we establish a one-to-one correspondence between regular 3-graphs, cyclic ADRGs of diameter 3 and certain rank 6 association schemes. As a consequence, we provide a new characterization of cyclic antipodal distance regular 3-fold covers of the complete graph in terms of association schemes. To keep the length of this paper reasonable we did not include all necessary preliminaries. In particular, we assume some knowledge of distance regular graphs, specifically, antipodal distance regular graphs of diameter 3. Also, we assume the reader is familiar with association schemes, in particular, the intersection algebra of an association scheme and its character-multiplicity table. An interested reader may find a more comprehensive consideration of all the diverse links exposed below as well as suggestions for further research in [7]. 2 Two-graphs and t-graphs 2.1 Two-graphs and regular two-graphs Two-graphs have roots originating in diverse areas of combinatorics, geometry and group theory, thus leading to different manifestations in the literature, such as: switching classes of graphs, sets of equidistant points in elliptic geometry, sets of equiangular lines in Euclidean geometry, binary maps of triples with vanishing coboundary, and double coverings of complete graphs (see the celebrated survey [12]). Our focus will be on the last two interpretations and the connection between them. We start with the classical definition and the classical viewpoint of two-graphs as switching classes of simple graphs. Let X be a set of n elements called vertices. For m G N denote by X{m} the set of all m-subsets of X. Definition 2.1. A set A C X{3} is a two-graph if every 4-subset of X contains an even (G {0,2,4}) number of members of A. Typically we use the notation (X, A) for a two-graph, and call A the set of odd triples. Definition 2.2. A two-graph (X, A) is called regular if every 2-subset {x, y} G X{2} is contained in the same number of triples from A. The most famous view of two-graphs is related to a special equivalence relation that is defined on the set of simple (undirected, no loops) graphs. First we remind the reader how to get a two-graph from a graph: Let r = (V, E) be a simple graph. The set of triples {u, v, w} of vertices, such that the induced subgraph r|{„jVjW} has an odd number of edges, forms a two-graph. Next, to define the equivalence relation we consider the operation of switching a graph with respect to a set of vertices. Definition 2.3. Let X C V be a subset of vertices of a simple graph r = (V, E). Switching with respect to X means interchanging the adjacencies and non-adjacencies between X and its complement V \ X. As a more appropriate setting to work with the operation of switching, J. J. Seidel proposed an alternative matrix representation of a simple graph: Definition 2.4. The Seidel adjacency matrix S = (si,j) of a graph r = (V, E) is a {0, -1,1}-matrix having: f 0 i = j, si,j = <-1 {i, j} e E, h {i,j}eE. In this notation, if the graph T' is obtained from r by switching with respect to X C V, then its Seidel adjacency matrix S' is obtained from S via a similarity transformation by a diagonal matrix having {- 1, 1} on its diagonal. Explicitly: S' = DSD, where Di,i = -1 ^^ i e X. As was implied above, switching is an equivalence relation on the set of all simple graphs of order n sharing the same prescribed vertex set. Furthermore we note that switching equivalent graphs give rise to the same two-graph, and have the same Seidel spectrum, thus allowing us to define the eigenvalues and their multiplicities of a two-graph. To sum up we have: Theorem 2.5. There is a 1-1 correspondence between two-graphs and switching classes of graphs. Theorem 2.6. A two-graph is regular if and only if it has two distinct (Seidel) eigenvalues pi > 0 > p2, such that pip2 = 1 - |X|. The following is an alternative definition of a two-graph. We call it the cohomological definition for reasons that will be clear soon. Definition 2.7. Let U2 be the group of square roots of unity. A set A C X{3} is a two-graph if the function: f : X{3} U2 defined by f (x) = -1 ^^ x e A, satisfies: f ({x,y,z}) ■ f ({x,y,t}) ■ f ({x,z,t}) ■ f ({y, z,t}) = 1 for any {x,y,z,t} e X{4}. Functions satisfying the equation in the above definition are called 3-cocycles (see below). It is clear that the two definitions are equivalent. Furthermore, we may refer to either (X, A), A or the function f as the two-graph. 2.2 The connection with double covers of complete graphs Two-graphs were originally introduced by Graham Higman to study 2-transitive representations of certain sporadic groups, in his description he used antipodal 2-fold covers of complete graphs. In [16], Taylor and Levingston established a one-to-one correspondence between two-graphs and antipodal 2-fold covers of complete graphs. This correspondence will be described in a more general setting with all details in the next section. Meanwhile we give an overview for the case of two-graphs. Let r be a graph with n vertices in the switching class of the two-graph f and let Sr be the Seidel adjacency matrix of r. Then by inserting a 2 x 2 matrix in the place of each entry of Sr according to the following rule: we obtain a 2n x 2n {0,1}-matrix which is the usual adjacency matrix of the corresponding 2-fold cover of Kn. The converse construction is done in a similar manner: substituting each 2 x 2 block of the adjacency matrix of a 2-fold cover of Kn (writing it in a suitable ordering of the vertices) with an element of {0,1, -1}. A 2-fold cover of Kn, when it is also distance regular, is called a Taylor graph, these are distance regular graphs with intersection array In the above mentioned correspondence, Taylor graphs correspond to regular two-graphs. This will be a particular case of our more general result later on. 2.3 Generalizing two-graphs Considering the cohomological definition of two-graphs, two very natural generalizations arise: • t-cocycles into U2 (functions f : —> U2 with a similar property as for two-graphs); • 3-cocycles into Ut (functions f : X{3} —> Ut, where Ut is the group of t-th roots of unity). Historically, the first of these was indeed the first to be considered. The first appearance of the term t-graph as a t-cocycle over U2 is due to D. Higman's generalization (see [4]) of E. Shult's graph extension theorem (see [13]). Other sources of this (design theoretical) generalization can be found in Mielants [11] or in [1]. In this case, a regular t-graph is a t-cocycle into U2 which is also a t-design. Here just few examples are known: regular 3-graphs on 8 and 12 points and a regular 5-graph on 12 points (see [10]). Our interest in the current presentation is the second way to generalize two-graphs, i.e. 3-cocycles into Ut. This direction was examined by D. Higman, and the main source of this is [5]. We begin with introducing some very basic elements of cohomology theory, in which terms t-graphs are defined. {k, 1; 1, n, k} . 2.4 Some cohomology Let X be a finite set with \X| = n. Let Z be a primitive root of unity of order t, and let Ut = < Z > denote the cyclic group of tth roots of unity generated by Z. Let x = (xi, x2,..., xp) e Xp. A function f : Xp —> Ut is called ap-cochain if: (i) f (x) = 1 (the identity element of Ut) for all x e Xp such that xi = xj for some 1 < i = j < P, (ii) if y results from x by interchanging xj and xj for some 1 < i = j < p then f (y) = (f (x))-1. The set of all p-cochains together with pointwise multiplication forms a group denoted by Cp (X, Ut ). Define the coboundary operator: Ö : Cp(X, Ut) Cp+1(X, Ut) by p f (x) = n (%)) i=0 where xc'i e Xp is obtained from x e Xp+1 by deleting the ith coordinate xi, and a is the inverse operation of Ut. For e e X and p > 1 we have the group homomorphism Ae : Cp(X,Ut) Cp-1(X,Ut) defined by Aef (x) = f (e, x) for x e Xp-1. Define the set of p-coboundaries: Bp(X,Ut) = {f |f e Cp-1(X,Ut)} , and the set of p-cocycles: Zp(X,Ut) = {f e Cp(X,Ut) \ Jf = 1} . Here 1 is the identity cochain in Cp+1 (X, Ut). It is routine to check that J2f = 1 for any (p - 1)-cochain f, and thus the coboundary of any (p - 1)-cochain is a p-cocycle. Two (p -1) -cochains have the same p-cocycle as their coboundary if and only if their quotient is a (p - 1)-cocycle. Thus, p-cocycles correspond to cohomology classes of (p -1)-cochains, as a generalization of Seidel switching we call the cohomology classes switching classes. Along the considered generalization of two-graphs and regular two-graphs we define: Definition 2.8. A t-graph is a 3-cocycle into Ut. Definition 2.9. A t-graph is called regular if for every pair x, y e X, the number of z e X \ {x, y} such that f (x, y, z) = a depends only on a e Ut. This number is denoted m(a). It is easy to check that in case t = 2, the definition of a 3-cocycle into U2 is compatible with the characterization given in Definition 2.7, and that the above definition of regularity is compatible with Definition 2.2. 2.5 Weights According to Higman, a weight on X with values in Ut is a 2-cochain w e C2 (X, Ut), from this point onward we will call them simply weights. Thus t-graphs are the coboundaries of weights. A weight w can be represented as a n x n matrix W with entries from Ut where: (W )x,y = w(x,y). Then W has 1 on its diagonal, and W * = W, where W * is obtained from W by transposing and inverting each entry. We will investigate the matrix representation of a weight with much more detail in the next section where we will focus on the case t = 3. Another way to represent a weight is as an antipodal t-fold cover of Kn. Definition 2.10. Let w : X2 —> Ut be a weight on X. To each element x e X we associate t vertices xi, x2,..., xt and define a graph rw = (V, E) on the resulting set V of t|X | vertices by {xi,yj} e E w(x, y) = (j-i. The resulting graph is a t-fold cover of the complete graph Kn, and if w(x, y ) = Zi then the set of edges between xi , x2 , ... , xt and yi , y2 , ... , yt forms a perfect matching which is given by the ith power of the permutation matrix of (1,2,..., t). Permuting x1, x2,..., xt according to some power of the permutation (1,2,..., t) amounts to a change of w in its switching class. 3 Regular 3-graphs From now on we focus on the case t = 3. We will prove that the situation for regular 3-graphs generalizes the case of regular two-graphs. In particular, regular 3-graphs are in 1 - 1 correspondence with regular (cyclic) (n, 3, c2)-covers. 3.1 Main conventions Let w : X2 —> U3 be a weight on X and |X| = n. The coboundary Jw of w is a 3-graph $ e Z3(X, U3) on X. Assume that $ is regular. Recall that this means that for every pair x, y of distinct elements of X and a e U3, the number m(a) of z e X \ {x, y} such that $(x, y, z) = a is independent of the choice of x and y. Denote a := m(1), b := m(Z) = m(Z2). We call (n, a, b) the parameters of the regular 3-graph $. We obtain the first restriction on the parameters by simple counting. Fix two vertices x, y e X, then: |X\{x,y}| = |{z e X | $(x,y,z) = 1}| + |{z e X | $(x, y, z) = Z}| + I {z e X I $(x, y, z) = Z2}| . Thus we have n - 2 = a + 2b. The corresponding graph rw is a 3-fold cover of Kn with exactly 3 types of matchings between fibres: x y xi •-• yi x2 •- -• V2 x3 -• y 3 xi x2 x3 yi y2 y3 xi x2 x3 yi y2 y3 w(x, y) = 1 w(x, y) = C w(x, y) = Z2 Figure 1: Matchings between fibres of rw The following subsection serves to remind and fix notation about matrices over the integral group ring, which is the setting in which we characterize regular 3-graphs. 3.2 Matrices over group rings Let T be a finite group. The elements of the integral group ring Z [T] are expressions of the form J2ag g geT where ag G Z. The ring operations are: J2agg ) + I E bgg ) = J2(ag + bg^ w(x, z) • w(x, z) 1 E: z£X = E 1 = nI. zex For x = y we have: (W2 ) x,y = E (W )x,z • (W )z,y zex = E w(x, z) • w(z, y) zex = E öw(y,x,z) • w(x,y) zex = ( E ^w(y, x, z M • w(x, y) Vzex / = ((m(1) + 2)1 + m(Z)Z + m(Z2)C!) • w(x,y) = ((a + 2)1 + bC,C2) • w(x, y). Summing up we get Equation (4.1). Using: bZ, Z2 (W - 1) = ÒU3 J - 6U3Ì - b1(W - I) we get Equation (4.2). □ Proposition 4.2. Every regular 3-graph with parameters (n, a, b) defines a cyclic (n, 3, b)-cover. Proof. We prove that the matrix C = W - 1 is the Godsil-Hensel matrix of the cyclic cover rw. We use Equation (4.2) to prove that C satisfies the condition of Theorem 3.1. C2 = (W - 1)2 = W2 - 2W + 1 = W2 - 2(W - 1) - 1 = nI +((a + 2)1 + bZ,Z2)(W - I) - 2(W - 1) - I = (n - 1)1 +(a1 + bC,C!)(W - 1) = (n - 1)1 + (a - b)1(W - 1 ) + bUs( J - 1 ) = (n - 1)1 + (a - b)1C + bUs(J - 1 ). Plugging in the values C2 = b, ai = a, r = 3 we obtain Equation (3.1) in Theorem 3.1. □ The converse is proved similarly: Proposition 4.3. Every cyclic (n, 3, c2)-cover defines a regular 3-graph with parameters (n, ai, C2). Proof. Let A be the Godsil-Hensel matrix of a cyclic (n, 3, c2)-cover. We show that W = A +1 is the matrix representation of a weight w in the switching class of a regular 3-graph. We have: W2 = (A + I )2 = A2 + 2A + I = (n - 1)I + (n - 2 - rc2)A + c2T(J - I) + 2A + I = nI + (n - rc2)A + c2T(J - I) = nI + (n - rc2)(W - I) + c2T(J - I). For the values b = Ö2, a = ai, r = 3 W satisfies Equation (4.2) in Lemma 4.1. □ To complete the picture we prove: Proposition 4.4. There is a 1 - 1 correspondence between regular 3-graphs and cyclic (n, 3, c2)-covers. Proof. Let w and w' be weights into U3. All that needs to be shown is: 5w = 5w' ^^ rw = rw/. As was explained after Definition 2.10, the switching of a weight w is interpreted as a cyclic permutation within the fibres of the corresponding cover rw, thus switching equivalent weights yield isomorphic covers. The converse is straightforward. □ As a consequence, using Theorem 9.2 of Godsil and Hensel in [2], we obtain a restriction on the parameter set of a regular 3-graph. Corollary 4.5. If (n, a, b) are the parameters of a regular 3-graph then 3|n. Proof. Since rw is a cyclic (n, 3, b)-cover, then by Theorem 9.2 in [2] we have 3|n. □ 5 Higman's note: clarification and corrections 5.1 Regular 3-graphs and association schemes Higman's first step in [6] is to define rw = (V, E), an antipodal 3-fold cover of Kn, with fibre set X. He then constructs a rank 4 symmetric association scheme from rw, this association scheme is (in a different ordering than the one that appears in [6]) the metric association scheme of the ADRG rw. Higman's key observation is the fact that this rank 4 association scheme admits a rank 6 fission by orienting all the non-edges of rw. We present this construction. Construction 5.1. Define: Ro =Idv, Rl = {(xi,Xi+1 (mod 3}) I i = 1, 2, 3, x e X } , R2 = { (xi,xi+2 (mod 3}) 1 i = 1, 2, 3,x e X] , R3 = E, R4 = {(xi, yj) 1 i = 1, 2, 3, {xi+1 (mod 3},yj} e E} , R5 = {(xi,yj) 1 i = 1 2, 3, {xi+2 (mod 3} ,yj} e E} . Remark 5.2. Notice that the relations R1, R2, R4, R5 are anti-symmetric, R1 = R2 and R4 = R5; Also S1 = R1 U R2 is the "distance 3" relation and S3 = R4 U R5 is the "distance 2" relation with respect to rw = (V, E). Proposition 5.3 (Higman). Ae (r) := (V, {Ri}5= 0) is an association scheme. Proof. We calculate the intersection matrices of Ae (r). For example, we compute p44: let (xi, yj ) e R4 and suppose w(xi, yj ) = Z (we may assume so due to switching), thus j = i + 1 (mod 3). We count the number of zk e V such that (xi,zk) e R4 and (zk, yj ) e R4: there are 3 types of z e X which contain such a zk : k : w(x, z) = w(z, y) Z2 = Z k = i +1 (mod 3) k = i — 1 (mod 3) Jw(x,y,z) = Z • Z • Z2 = Z, > Jw(x, y, z) = Z • 1 • 1 = Z, Jw(x,y,z) = Z • Z2 • Z = Z. w(x, z) = 1 w(z,y) = 1 w(x, z) = Z w(z,y) = Z2 Thus Jw(x, y, z) = Z ^^ z is one of the above 3 types, hence p44 = b. In the same manner we obtain the intersection matrices {Bi}5=0: Bo = B3 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 ,B2 0 1 0 0 0 0 0 0 0 1 0 0 , B1 = 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 - 0 0 0 1 0 0" - 0 0 0 0 1 0" 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 , B4 0 0 0 1 0 0 n 1 0 0 a b b = 0 n 1 0 b a b , 0 n— 1 0 b a b 0 0 n — 1 b b a 0 0 n— 1 b b a n —1 0 0 a b b - 0 0 0 0 0 1" 0 0 0 1 0 0 B 0 0 0 0 1 0 □ 1 0 0 n b b a n —1 0 —0 a b b 0 n — 1 0 b a b It turns out that the existence of such a rank 6 association scheme is a sufficient condition: Proposition 5.4 (Higman). Every rank 6 association scheme with parameters as in the construction of A6(rw ) arises from a regular 3-graph. Proof. Let (V, {Ri}f=0) be an association scheme with parameters (and notation) as in Construction 5.1. Define T = R0 U Ri U R2, then T is an equivalence relation on V with equivalence classes of size 3. Denote X = V/T. We give a labeling of the elements of V as a 3-fold cover of K|X |, and then we verify that in this cover we only have matchings of the 3 types shown in Figure 1. Let a e X be any fibre, and label its elements by a i, a2, a3 so that (ai,a2) e Ri. Then (o^ ai+1 (mo d 3)) e Ri for i = 1, 2, 3. We now label the elements of each fibre x = a in X by xi, x2, x3 so that (ai, xi) e R3 for i = 1,2, 3. To prove that (V, R3) is a 3-fold cover of K|X| with matchings of the 3 permitted types, we prove two things: (1) (xi, xi+i (mod 3)) e Ri for all x e X and i = 1, 2, 3, (2) there is no matching such that (xi, yj) e R3 and (xj, yi) e R3, where i = j. Proof of (1): Assume that (xi, xi+i) e R2. Let k be such that (ai, xi+i) e Rk. Then we have: (xi, ai) e R3, (xi,xi+i) e R2, (xi+i, ai) e Rfc'. Therefore: pik' = 0. Examining column 3 in the matrix B2, we deduce that k' = 4, which means that k = 5. Also, we have: (ai, xi+i) e Rfc, (ai, ai+i) e Ri, (ai+i,xi+i) e R3. This implies that: Pk3 = 0. Examining row 3 in the matrix Bi, we deduce that k = 4, which is a contradiction. Proof of (2): Assume that (xi, yj) e R3 and (xj, yi) e R3 for some i = j. Let k be such that (xi, yi) e Rk. W.l.o.g we may assume that j = i + 1 (mod 3). Then we have: (xi,yi) e Rfc, (xi,xi+i) e Ri, (xi+i,yi) e R3. Therefore: pk3 = 0. Examining row 3 in the matrix Bi we see k = 4. Also, we have: (yi, xi) e Rfc', (yi,yi+i) e Ri, (yi+i,xi) e R3. Thus we obtain: Pi 3 = 0, which implies that k' = 4 and k = 5, a contradiction. It follows that all the matchings of the graph (V, R3) are of the 3 types shown in Figure 1, and we can define a weight w on X by w(x, x) = 1 and w(x, y) = 1, Z or Z2 for x = y according to as the matching is of the first, second or third type. It is straightforward to verify that 5w is regular. □ 5.2 Characterization and feasibility conditions We now sum up the results of the previous sections with our characterization of regular 3-graphs. Notice that the equivalence of (ii) and (iii) is a characterization of cyclic (n, 3, c2 )-covers in terms of association schemes. Corollary 5.5. Let r be an antipodal 3-fold cover of Kn. The following are equivalent: (i) r defines a regular 3-graph with parameters (n, a, b); (ii) r is a cyclic (n, 3, b)-cover; (iii) A6(r) is an association scheme. Using this characterization we would like to obtain feasibility restrictions on the parameters (n, a, b) of regular 3-graphs. We begin by calculating the character-multiplicity tables of A4(rw) and A6(rw). We used the well-known computer software Mathematica to calculate these tables, the program code is presented in [7]. The character-multiplicity table of A4 (rw) is: 1 2 n — 1 2(n — 1)" 1 1 2 —1 —2 n — 1 1 —1 a —a zi 1 —1 ß —ß z2 Here: • a and ß are the roots of x2 — (a — b)x — (n — 1) = 0, • zi = Jnü, 1 ß—a ' • Z2 = 2n — zi = O—ß. If z1 = z2 = n then we have a = —ß, and a, ß = ±Vn — 1. Otherwise, z2 — zi is a non-zero integer, and we have: o , - ß A 0 ( a + ß Z2 — zi = 2n -- — —- = 2n a — ß ß — a a — ß This means that a — ß = \J(a — b)2 + 4(n — 1) is rational, i.e. (a — b)2 + 4(n — 1) is a square, which implies that a and ß are rational algebraic integers, and thus are integers. The character-multiplicity table of A6(Tw ) is: 1 1 1 n — 1 n — 1 n — 1 1 Z Z2 a aZ aZ2 1 Z2 Z a aZ2 aZ 1 1 1 —1 —1 —1 1 Z Z2 ß ßZ ß to 1 Z2 Z ß ß to ßZ 1 zi/2 zi/2 n — 1 Z2/2 _Z2/2_ Here: • aZ and aZ2 are the roots of x2 + ax + a2 = 0, • ßZ and ßZ2 are the roots of x2 + ßx + ß2 = 0. Remark 5.6. In Higman's note appeared the equations: • x2 - ax + (^^ + a2) = 0, • x2 - ßx + (+ ß2) = 0, which led him to the false conclusion that n must be odd. These equations are the result of a miscalculation of the intersection matrices B3 and B5 of A6 (Tw ) (compare these matrices from our paper with those from the note [6]). Summing up all the considered restrictions we obtain: Proposition 5.7. Necessary conditions for the set (n, a, b) of parameters of a regular 3-graph are: (i) n = a + 2b + 2, (ii) 3|n, (iii) The roots a and ß of the equation x2 — (a — b)x — (n — 1) = 0 are integers, (iv) a - ß divides na. Proof. Item (i) appears in the beginning of Section 3. Item (ii) is Corollary 4.5. Item (iii) comes from the latter analysis of the character-multiplicity table of A4(rw), and (iv) is just the integrality of the multiplicity = a— of A6 (rw ). □ These feasibility conditions provide a list of just 64 feasible parameter sets with n < 1000. We refer to [7] for the complete list and details about known constructions for some of them. 5.3 The symplectic example D. G. Higman provided an infinite family of regular 3-graphs which is described briefly below. In [5], Higman considers a more general cohomological setting, and presents several group theoretic examples of regular 3-cocycles (here cochains are functions into a monoid with the appropriate conditions). These examples are mainly extensions of examples by D. E. Taylor in [15]. We mention one of them. In this example, we consider weights with values in the additive group of the field GF(q), thus we will use additive notation: C+ instead of C2, J+ instead of J. etc. Let V be a 2m-dimensional vector space over GF(q). Let B be a non-degenerate alternating bilinear form on V. Then B e C+ (V, GF(q)) is a weight on V with values in GF(q). In case q is a prime, this 3-cocycle is a q-graph. To see that in this case $ = J+B is a regular q-graph we consider the symplectic group Sp(2m, q). It acts transitively on the non-zero vectors of V, thus the subgroup H := VSp(2m, q) of the affine group on V acts 2-transitively on the vectors of V. The coboundary $ = J+B is invariant under translations and is therefore invariant under the action of H on V. This provides an infinite family of regular q-graphs for every prime q. 6 New constructions The equivalence of regular 3-graphs with parameters (n, a, b) and cyclic (n, 3, b)-covers provides a rich source of new examples of regular 3-graphs. In their recent paper [8], Klin and Pech present a construction of cyclic (m2,3, m2)-covers from generalized Hadamard matrices of order m over the cyclic group of order 3; the set of such matrices is denoted by gH(U3, m). Their method takes as input any generalized Hadamard matrix H e gH(U3, m) and produces a so-called skew generalized Hadamard matrix W e gH(U3, m2) of order m2; such matrices correspond to cyclic (m2,3, mr)-covers, this is the Godsil-Hensel matrix of the cover. We used classifications of generalized Hadamard matrices with suitable parameters (see [3], [9] and [14]) to construct all the corresponding non-isomorphic cyclic covers using the Klin-Pech method, which provide different regular 3-graphs. A summary of our new constructions of regular 3-graphs: • 1 new example with parameters (36,10,12), • 1 new example with parameters (45,19,12) (exceptional), • 1 new example with parameters (81,25,27), • 1 new example with parameters (144,46,48), • 28 new examples with parameters (324,106,108). For the complete list of feasible parameter sets with n < 1000, and details about the above examples see [7]. 7 Extension to regular t-graphs with t > 4 The theory outlined in this paper can be extended to regular t-graphs with any t > 4 only if we impose certain restrictions on the parameters of the regular t-graph. For example, when t is odd, the parameters of a regular t-graph are: n, m(1), m(Z ) = m(Zt-i ), m(Z t-1/2)= m(Zt+1/2). A graph rw defined by a regular t-graph will be distance regular only if most parameters of the regular t-graph are equal. Explicitly, in the case that t is odd we demand: m(Z ) = m(Z2 ) = ••• = m(Zt-1/2). In this case, these will also be cyclic covers since for any t we have Cst (Ct) = Ct. Here we use the notation Ct < St for the cyclic group Ct = ((1,2,..., t)). Higman's theory also extends to regular t-graphs with t > 4 in the case of equal parameters (as described above). The construction of A4(rw ) is exactly the same, and it has a rank 2t refinement which completely determines the weight w (analogously to A6 (rw ) in the case of regular 3-graphs). Thus, the extension of our theory to t > 4 is described schematically in Figure 2: Regular t-graph ^^ (n, t, c2)-cover with-^ The metric association with equal cyclic matchings scheme of the cover parameters Association scheme with 2t - 1 classes and particular parameters Figure 2: Extension to t > 4 Acknowledgments The content of this paper is part of my MSc thesis completed in 2011 under the supervision of Misha Klin to whom I am thankful for his guiding hand. I thank Misha Muzychuk and Eran Nevo who were members of the committee for my thesis defence and gave helpful remarks. I thank Christian Pech whose valuable remarks on my thesis are reflected quite essentially in this paper. Finally, I thank the anonymous referees for their helpful remarks. References [1] P. J. Cameron and J. H. van Lint, Graphs, codes and designs, volume 43 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 1980, revised edition of Graph theory, coding theory and block designs, an older version of Designs, Graphs, Codes and their Links. [2] C. D. Godsil and A. D. Hensel, Distance regular covers of the complete graph, J. Combin. Theory Ser. B 56 (1992), 205-238. [3] M. Harada, C. Lam, A. Munemasa and V. D. Tonchev, Classification of generalized Hadamard matrices H(6, 3) and quaternary Hermitian self-dual codes of length 18, Electron. J. Combin. 17 (2010), #R171. [4] D. G. Higman, Remark on Shult's graph extension theorem, in: Finite groups '72 (Proc. Gainesville Conf., Univ. Florida, Gainesville, Fla., 1972), North-Holland Math. Studies, Vol. 7, North-Holland, Amsterdam, 1973, pp. 80-83. [5] D. G. Higman, Weights and t-graphs, Bull. Soc. Math. Belg. Sér. A 42 (1990), 501-521. [6] D. G. Higman, A note on regular 3-graphs, 1994, unpublished draft. [7] D. Kalmanovich, Regular t-graphs, antipodal distance regular graphs of diameter 3 and related combinatorial structures, MSc thesis, Ben-Gurion University of the Negev, 2011, http: //www.math.bgu.ac.il/~dannykal. [8] M. Klin and C. Pech, A new construction of antipodal distance regular covers of complete graphs through the use of Godsil-Hensel matrices, Ars Math. Contemp. 4 (2011), 205-243. [9] V. Mavron and V. Tonchev, On symmetric nets and generalized Hadamard matrices from affine designs, Journal of Geometry 67 (2000), 180-187. [10] W. Mielants, A regular 5-graph, AttiAccad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 60 (1976), 573-578. [11] W. Mielants, Remark on the generalized graph extension theorem, European J. Combin. 1 (1980), 155-161. [12] J.J.Seidel, A survey of two-graphs, in: Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo I, pages 481-511, Atti dei Convegni Lincei, No. 17. Accad. Naz. Lincei, Rome, 1976. [13] E. Shult, The graph extension theorem, Proceedings of the American Mathematical Society, 33 (1972), 278-284. [14] C. Suetake, The classification of symmetric transversal designs STD4[ 12; 3]s, Designs, Codes and Cryptography 37 (2005), 293-304. [15] D. E. Taylor, Regular 2-graphs, Proc. London Math. Soc. 35 (1977), 257-274. [16] D. E. Taylor and R. Levingston, Distance-regular graphs, in: Combinatorial mathematics (Proc. Internat. Conf. Combinatorial Theory, Australian Nat. Univ., Canberra, 1977), volume 686 of Lecture Notes in Math., Springer, Berlin, 1978, pp. 313-323. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 117-125 Some properties of the Zagreb eccentricity indices Kinkar Ch. Das * Department of Mathematics, Sungkyunkwan University Suwon 440-746, Republic of Korea Dae-Won Lee Sungkyunkwan University, Suwon 440-746, Republic of Korea Ante Graovac Faculty of Science, University of Split, Nikole Tesle 12, HR-21000 Split, Croatia Received 10 October 2011, accepted 20 January 2012, published online 5 June 2012 Abstract The concept of Zagreb eccentricity (E1 and E2) indices was introduced in the chemical graph theory very recently [5,12]. The first Zagreb eccentricity (Ei) and the second Zagreb eccentricity (E2) indices of a graph G are defined as Ei = Ei(G)= £ e, ViEV (G) and E, = E2(G)= £ ei • ej , vivj EE(G) where E(G) is the edge set and ež is the eccentricity of the vertex v in G. In this paper we give some lower and upper bounds on the first Zagreb eccentricity and the second Zagreb eccentricity indices of trees and graphs, and also characterize the extremal graphs. Keywords: Graph, first Zagreb eccentricity index, second Zagreb eccentricity index, diameter, eccentricity. Math. Subj. Class.: 05C40, 05C90 * Corresponding author. E-mail addresses: kinkardas2003@googlemail.com (Kinkar Ch. Das), haverd2001@gmail.com (Dae-Won Lee), ante.graovac@irb.hr (Ante Graovac) 1 Introduction Mathematical chemistry is a branch of theoretical chemistry using mathematical methods to discuss and predict molecular properties without necessarily referring to quantum mechanics [1, 8, 14]. Chemical graph theory is a branch of mathematical chemistry which applies graph theory in mathematical modeling of chemical phenomena [6]. This theory has an important effect on the development of the chemical sciences. Topological indices are numbers associated with chemical structures derived from their hydrogen-depleted graphs as a tool for compact and effective description of structural formulas which are used to study and predict the structure-property correlations of organic compounds. Molecular descriptors are playing significant role in chemistry, pharmacology, etc. Among them, topological indices have a prominent place [13]. One of the best known and widely used is the connectivity index, x, introduced in 1975 by Milan Randic [11]. The Randic index is one of the most famous molecular descriptors and the paper in which it is defined is cited more than 1000 times. The first Mi, and the second M2, Zagreb indices (see [2],[3],[4],[7],[9] and the references therein) are defined as: Mi = Mi (G) = £ d? vi ev (G) and M? = M?(G) = £ di ■ dj. vivj eE(G) where di is the degree of the vertex vi e V (G) in G. Let G = (V, E) be a connected simple graph with | V (G) | = n vertices and | E (G) | = m edges. Also let di be the degree of the vertex vi, i = 1,2,..., n. For vertices vi, v j e V (G), the distance dG(vi, v j ) is defined as the length of the shortest path between vi and vj in G. The eccentricity of a vertex is the maximum distance from it to any other vertex, ei = max dG(vi,vj). vj ev (G) The maximum eccentricity over all vertices of G is called the diameter of G and denoted by d. The invariants based on vertex eccentricities attracted some attention in Chmistry. In an analogy with the first and the second Zagreb indices, M. Ghorbani et al. and D. Vukicevic et al. define the first E1, and the second, E2, Zagreb eccentricity indices by [5, 12] Ei = Ei (G) = £ e? (1.1) viev (G) and E? = E?(G) = £ ei ■ ej. (1.2) vivj eE(G) where E(G) is the edge set and ei is the eccentricity of the vertex vi in G. Let G = (V (G), E (G)). If V (G) is the disjoint union of two nonempty sets Vi(G) and V2 (G) such that every vertex in Vi (G) has degree r and every vertex in V2 (G) has degree s, then G is (r, s)-semiregular graph. When r = s, is called a regular graph. As usual, Ka,b (a + b = n), Pn and K1,n-1 denote respectively the complete bipartite graph, the path and the star on n vertices. A vertex of a graph is said to be pendent if its neighborhood contains exactly one vertex. An edge of a graph is said to be pendent if one of its vertices is a pendent vertex. Now we calculate El(Pn) 112 (n — 1)(7n2 — 2n) if n is even I2 (n — 1)(7n2 — 2n — 3) if n is odd. (1.3) and E2(Pn) 12 n(7n2 — 21n + 20) if n is even 12 (n — 1)(7n2 — 14n + 3) if n is odd. (1.4) Also we have E1(K1,n_1) = 4n — 3 and E2(#1,n-1) = 2n — 2. Denote by Tn, is a tree of order n with maximum degree n — 2. We have E1(TT„) = 9n — 10, E2(Tn) = 6n — 8. In this paper we give some lower and upper bounds on the first Zagreb eccentricity and the second Zagreb eccentricity indices of trees and graphs, and also characterize the extremal graphs. 2 Lower and upper bounds on Zagreb eccentricity indices We now give lower and upper bounds on the Zagreb eccentricity indices of trees. Theorem 2.1. Let T be a tree with n vertices. Then (i) E1(K1,n_1) < E1(T) < E1(Pn) (2.1) and (ii) E2(K1,n_1) < E2(T) < E2(Pn). (2.2) Moreover, the left hand side (right hand side, respectively) equality holds in (2.1) and (2.2) if and only if G = K1,n-1 (G = Pn, respectively). Proof. Upper bound: If T is isomorphic to path Pn, then the right hand side equality holds in (2.1) and (2.2). Otherwise, T ^ Pn. Let d be the diameter of tree T. Then there exists a path Pd+1 : v1v2... vd+1 of length d in T. Thus we have the eccentricity of a vertex vi in tree T, ei = max|dG(vj, V1 ), dG(vi,vd+1)} . Since T is a tree, both vertices v1 and vd+1 are pendent vertices. Thus we have ei < d for each vi g V (G). Since T ^ Pn, let vk (k = 1, d + 1) be a vertex of degree one, adjacent to vertex vj in T. We transform T into another tree T * by deleting the edge vk vj and join the vertices vd+1 and vk by an edge. Then the longest path Pd+2 : v1v2... vd+1vk of length d + 1 in T *. Let the vertex eccentricities be e1, e2,..., en in T *. Therefore we have e* = max{dG(vt, v1), d*G(vt,vk)} = max{dG(vt, v1), dG(vt, vd+1) + 1} > max{dG(vt, v1), dG(vt,vd+1)} = et (as d*G(vt,vk) = dG(vt,vd+1) + 1) for t = k whereas e*k = d +1 > d > ek (dG (vi, vj ) is the length of the shortest path between vertices vi and vj in T*). So we have e* e* > er es for vrvs = vkvj, vkvd+1 and e*k e*d+1 d(d +1) > d2 > ek e j. Using above result we get E1(T*) - Ei(T) = V e*2 - V e,2 > ek2 - e\ > 0 _y i _y ViEV (T *) ViEV (T ) and E2(T*) - E2(T)= V e* e* - V er es > e* ed+1 - ek ej > 0. vr vsEE(T*) vr vsEE(T) Therefore we have Ei(T*) >Ei(T), i = 1, 2. By the above described construction we have increased the value of Ei(T), i = 1,2. If T * is the path, we are done. If not, then we continue the construction as follows. Next we choose one pendent vertex (= v1, vk) from T*, etc. Repeating the procedure sufficient number of times, we arrive at a tree in which the maximum degree 2, that is, we arrive at path Pn. Lower bound: If T is isomorphic to star K1n-1, then the left hand side equality holds in (2.1) and (2.2). If T is isomorphic to Tn, then the left hand side inequality is strict in (2.1) and (2.2). Otherwise, T ^ K1jn-1, Tn . Suppose that a path Pd+1 : v1v2... vd+1 of length d in T, where d is the diameter of T. Without loss of generality, we can assume that d2 > dd (the degree of vertex v2 is greater than or equal to the degree of vertex vd). Now choose vi to be an arbitrary maximum degree vertex, unless vd has maximum degree, in which case vi is chosen to be v2. We transform T into another tree T by deleting the edge vd vd+1 and join the vertices vi and vd+1 by an edge. Let the vertex eccentricities be e1, e2,..., en in tree TT. Similarly, as before we obtain et < et for all t = 1, 2,..., n. Using above we get E1(T) - E1(T)= V T2 - V e,2 < 0 vieV(T) viEV(T) and E2(!T) - E2(T)= V Tr Ts - V er es < 0. vr vsEE(T) vr vsEE(T) Therefore we have Ei(T) < Ei(T), i = 1,2. By the above described construction we have non-increased the value of Ei(T), i = 1,2. If T is to the tree Tn , we are done. If not, then we continue the construction as follows. Next we choose one pendent vertex from longest path in T such that its adjacent vertex is not maximum degree vertex. Now we delete that pendent edge and join the pendent vertex to the maximum degree vertex, etc. Repeating the procedure sufficient number of times, we arrive at a tree in which the maximum degree n - 2, that is, we arrive at tree Tn. This completes the proof. □ We now give lower and upper bounds on the Zagreb eccentricity indices of bipartite graph. Theorem 2.2. Let G be a connected bipartite graph of order n with bipartition V(G) = U U W, U n W = 0, | U| = p and |W| = q. Then (i) Ei(Kp,q) < Ei(G) < E1(Pn) (2.3) and (ii) E2(Kp,q) < E2(G) < E2(Pn). (2.4) Moreover, the left hand side (right hand side, respectively) equality holds in (2.3) and (2.4) if and only if G = Kp,q (G = Pn, respectively). Proof. If G is isomorphic to a complete bipartite graph Kp q, then the left hand side equality holds in (2.3) and (2.4). Otherwise, G £ Kp,q. If we add an edge in G, then each vertex eccentricity will non-increase. Thus we have e, (G + e) < e,(G). Using this property, one can see easily that E1(G) > E1(Kp,q\{e}) > E1(Kp,q) and E2(G) > E2(Kpq\{e}) > E2(Kpq), where e is any edge in Kpq. Let T be a spanning tree of connected bipartite graph G. Then by the above property, Ei(G) < Ei(T) and E2(G) < E2(T). Using this result with Theorem 2.1, we get the right hand side inequality in (2.3) and (2.4). Moreover, the right hand side equality holds in (2.3) and (2.4) if and only if G = Pn. This completes the proof. □ In [10], Hua et al. proved the following result in Theorem 3.1. Lemma 2.3. Let G be a connected graph with e, = n - d, for any vertex v, G V (G). If G £ P4, then e, < 2 for any vertex v, G V (G). We now give some relation between first Zagreb index and the first Zagreb eccentricity index of graphs. Theorem 2.4. Let G be a connected graph of order n with m edges. Then E1(G) < M1(G) - 4mn + n3, (2.5) where M1 (G) is the first Zagreb index in G. Moreover, the equality holds in (2.5) if and only if G = P4 or G = Kn or G is isomorphic to a (n - 1, n - 2)-semiregular graph. Proof. If G = P4, then the equality holds in (2.5). Otherwise, G £ P4. Now, E1(G) = e2 < (n - d,)2 as e, < n - d VieV(O) ViEV (G) = M1(G) - 4mn + n3 as M1(G) = ^ d2, ^ d, = 2m. ViEV (G) ViEV (G) First part of the proof is over. Now suppose that equality holds in (2.5). Then e, = n - d, for all v G V (G). By Lemma 2.3, we conclude that e, < 2 for any vertex v, G V (G) as G £ P4. Since e, = n - di for any vertex v, G V (G), we must have di = n - 1 or n - 2 for any vertex v, G V (G), that is, G = Kn or G is isomorphic to a (n - 1, n - 2)-semiregular graph. Conversely, one can see easily that (2.5) holds for P4 or Kn or (n -1, n - 2)-semiregular graph. □ Remark 2.5. (n - 1, n - 2)-semiregular graph is obtained by deleting i independent edges from Kn, 1 < i < L f J. We now give some relation between first Zagreb index, second Zagreb index and the second Zagreb eccentricity index of graphs. Theorem 2.6. Let G be a connected graph of order n with m edges. Then E2 (G) < M2(G) - nMi (G) + mn2, (2.6) where Mi (G) is the first Zagreb index, M2 (G) is the second Zagreb index in G. Moreover, the equality holds in (2.6) if and only if G = P4 or G = Kn or G is isomorphic to a (n - 1, n - 2)-semiregular graph. Proof. Now, E2(G)= E ei • ej vivj eE(G) <53 (n - di)(n - dj) as ej < n - di and ej < n - dj vivj eE(G) = E (n2 + dj dj - (di + dj )nj vivj eE(G) = M2(G) - nMi(G)+ mn2. First part of the proof is over. Moreover, one can see easily that the equality holds in (2.6) if and only if G = P4 or G = Kn or G is isomorphic to a (n - 1, n - 2)-semiregular graph, by the proof of Theorem 2.4. □ (i Figure 1: Graphs G* and G**. Let K2,a_2 be a connected graph of order a obtained from the complete bipartite graph K2,a—2 with the vertices of degree a - 2 are adjacent. Denote by G*, is a connected graph of order n, obtained from K2 _2 2 by attaching two paths Pq+1 to two of its vertices of degree n - 2q - 1. Let r1 be the class of graphs H1 = (V, E) such that H1 is connected graph of diameter d (d = 2q +1) with V (G*) = V (H1) and E(G*) C E (H1). Let K|,a_2 be a connected graph of order a +1 obtained from the complete bipartite graph K2,a_2 with the vertices of degree a - 2 are adjacent to a new vertex. Denote by G**, is a connected graph of order n, obtained from K|,n_2q_1 by attaching two paths Pq to two of its vertices of degree n - 2q. Let r2 be the class of graphs H2 = (V, E) such that H2 is connected graph of diameter d (d = 2q + 2) with V (G*) = V (H2) and E (G*) C E(H2 ). We now give another lower bound on E1(G) in terms of n, d and also characterize the extremal graphs. Theorem 2.7. Let G be a connected graph of order n with diameter d. Then fl2 (3nd2 +6nd + 3n + 4d3 + 3d2 - 4d - 3) if d +1 is even ) ) >\ 12 (3nd + 4d2 +9d + 2) if d +1 is odd with equality holding if and only if G = Pn or G G r1 or G G r2. Proof. Since G has diameter d, G contains a path Pd+1: v1 v2 ..., vd+1. Moreover, n > d +1 and ei > [ f ], i = 1, 2,... , n. If n = d +1, then G = Pn and the equality holds in (2.7). Otherwise, n > d + 1. By (1.3), we get d+1 (7d2 + 12d + 5) if d +1 is even Ve2 = 12 (7d2 + 12d + 5) 11 d + 1 (2.8) j (7d2 + 12d + 2) if d +1 is odd. Since e, > , i = 1, 2,... , n, using above result, we get d+1 n E1(G) = £ e? + £ i=1 i=d+2 12 (7d2 + 12d + 5) + (n - d - 1)[2 ]2 if d +1 is even 12 (7d2 + 12d + 2) + 4 (n - d - 1)d2 if d + 1 is odd, from which we get the required result (2.7). First part of the proof is over. Now suppose that equality holds in (2.7) with n > d +1. From equality in (2.9), we get d for i = d + 2 , d + 3 , . . . , n. 2 Using above result we conclude that all the vertices vd+2, vd+3 ,... , vn_1 and vn are adjacent to vertices vq and vq+2 (when d = 2q), or vq+1 and vq+2 (when d = 2q + 1). Hence G G r1 or G G r2. Conversely, one can see easily that (2.7) holds for path Pn or graph G, where G G r1 or G G r2. □ We now give another lower bound on E2(G) in terms of m, d and also characterize the extremal graphs. Theorem 2.8. Let G be a connected graph of order n with diameter d. Then /12 (3md2 + 6md + 4d3 - 6d2 - 4d + 3m + 6) if d +1 is even ) T2 (3md + 4d2 - 4) if d +1 is odd (2.10) with equality holding if and only if G = Pn or G G ri or G G r2. Proof. By (1.3), we get V _/t+T (7d2 - 7d +6) if d +1 is even JtP ) ei ej 112 (7d2 - 4) if d +1 isodd. (. ) Vi v j eE(Pd+1) K 12 K ' Since e, > , i = 1, 2,... , n, we have E2(G)= ei ej + XI ei ej ViVj £E(Pd+i) ViVj £E(G\Pd+i) f^+i (7d2 - 7d + 6) +(m - d) \d]2 if d +1 is even "j 12 (7d2 - 4) + i (m - d)d2 if d +1 isodd, from which we get the required result (2.10). Rest of the proof is similar as Theorem 2.7. □ Acknowledgement. The authors are grateful to the two anonymous referees for their careful reading of this paper and strict criticisms, constructive corrections and valuable comments on this paper, which have considerably improved the presentation of this paper. The first author's research is supported by Sungkyunkwan University BK21 Project, BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea. References [1] S. J. Cyvin and I. Gutman, Kekule Structures in Benzenoid Hydrocarbons, Lecture Notes in Chemistry, Vol 46, Springer Verlag, Berlin, 1988. [2] K. C. Das, Maximizing the sum of the squares of the degrees of a graph, Discrete Math. 285 (2004), 57-66. [3] K. C. Das and I. Gutman, Some Properties of the Second Zagreb Index, MATCH Commun. Math. Comput. Chem. 52 (2004), 103-112. [4] K. C. Das, I. Gutman and B. Zhou, New upper bounds on Zagreb indices, J. Math. Chem. 46 (2009), 514-521. [5] M. Ghorbani and M. A. Hosseinzadeh, A new version of Zagreb indices, Filomat 26 (2012), 93-100. [6] A. Graovac, I. Gutman and N. Trinajstic, Topological Approach to the Chemistry of Conjugated Molecules, Springer Verlag, Berlin, 1977. [7] I. Gutman and K. C. Das, The first Zagreb indices 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), 83-92. [8] I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer Verlag, Berlin, 1986. [9] I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. III. Total n-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535-538. [10] H. Hua and S. Zhang, Relations between Zagreb coindices and some distance-Based topological indices, MATCH Commun. Math. Comput. Chem. 68 (2012), 199-208. [11] M. Randic, On characterization of molecular branching, J. Am. Chem. Soc. 97 (1975), 66096615. [12] D. Vukicevic and A. Graovac, Note on the comparison of the first and second normalized Zagreb eccentricity indices, Acta Chim. Slov. 57 (2010), 524-528. [13] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2000. [14] N. Trinajstic and I. Gutman, Mathematical chemistry, Croat. Chem. Acta 75 (2002), 329-356. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 127-145 Consensus strategies for signed profiles on graphs Kannan Balakrishnan Department of Computer Applications, Cochin University of Science and Technology Cochin - 682 022, India Manoj Changat * Department of Futures Studies, University of Kerala, Trivandrum - 695 034, India Henry Martyn Mulder t Econometrisch Instituut, Erasmus Universiteit P. O. Box 1738, 3000 DR Rotterdam, Netherlands Ajitha R. Subhamathi Department of Computer Applications, N.S.S. College, Rajakumari, Idukki, Kerala, India Received 18 October 2011, accepted 28 March 2012, published online 15 June 2012 Abstract The median problem is a classical problem in Location Theory: one searches for a location that minimizes the average distance to the sites of the clients. This is for desired facilities as a distribution center for a set of warehouses. More recently, for obnoxious facilities, the antimedian was studied. Here one maximizes the average distance to the clients. In this paper the mixed case is studied. Clients are represented by a profile, which is a sequence of vertices with repetitions allowed. In a signed profile each element is provided with a sign from {+, -}. Thus one can take into account whether the client prefers the facility (with a + sign) or rejects it (with a - sign). The graphs for which all median sets, or all antimedian sets, are connected are characterized. Various consensus strategies for signed profiles are studied, amongst which Majority, Plurality and Scarcity. Hypercubes are the only graphs on which Majority produces the median set for all signed profiles. Finally, the antimedian sets are found by the Scarcity Strategy on e.g. Hamming graphs, Johnson graphs and halfcubes. Keywords: Plurality strategy, median, majority rule, Hamming graph, Johnson graph, halfcube. Math. Subj. Class.: 05C99, 05C12, 90B80 * Research work is supported by NBHM/DAE under grant No.2/48(2)/2010/NBHM-R & D t Research work was carried out when this author visited University of Kerala, under the Erudite Scheme of the Government of Kerala, January 3-16, 2011 E-mail addresses: bkannan@cusat.ac.in (Kannan Balakrishnan), mchangat@gmail.com (Manoj Changat), hmmulder@few.eur.nl (Henry Martyn Mulder), ar.subhamathi@gmail.com (Ajitha R. Subhamathi) 1 Introduction Most of the facility location problems in the literature are concerned with finding locations for desirable facilities. The goal there is to minimize a distance function between facilities and the demand sites (clients). One way to model this is using a network, see for instance [23, 24, 17]. In the discrete case one uses graphs, and clients and facilities are to be positioned on vertices. One may formulate such location problems also in terms of achieving a consensus amongst the clients. Thus it becomes a problem in Consensus Theory. This approach has been fruitful in many other applications, e.g. in social choice theory, clustering, and mathematical biology, see for instance [7, 16, 15, 22]. From the view point of median consensus the classical result of Goldman [12] is very interesting: to find the median in a tree, just move to the majority of the clients. In [20], this majority strategy was formulated for arbitrary graphs. It was proved that majority strategy finds all medians for any set of clients if and only if the graph is a so-called median graph. Clients are termed as profiles in the language of graph theory, defined as a sequence of vertices in which vertices are allowed to repeat. The class of median graphs comprises that of the trees as well as that of the hypercubes and grids. It allows a rich structure theory [18, 13, 21] and has many and diverse applications, see, for. e.g., [14], for median type consensus. In the majority strategy we compare the two ends of an edge v and w: if we are at v and at least half of the clients are strictly nearer to w than to v, then we move to w. One could relax the requirement for making a move as follows: one may move to w if there are at least as many clients closer to w than to v. Note that in the latter case less than half may actually be closer to w because there are many clients having equal distance to v and w. This idea of relaxing the majority strategy is formalized as plurality strategy in [4]. Other consensus strategies known as Condorcet, hill climbing and Steepest ascent hill climbing strategies were also proposed in [4]. There it is proved that the plurality, hill climbing and steepest ascent hill climbing strategies starting at an arbitrary vertex for arbitrary profiles will always return the median set of the profile if and only if the graph has connected medians. However just as important are the problems dealing with the location of undesirable or obnoxious facilities, such as nuclear reactors, garbage dumps or water purification plants, see [9, 10, 11]. Here the criterion for optimality is maximizing the sum of the distances from the location of the obnoxious facility to the locations of the clients. The problem is known as the antimedian problem. In general any two subgraphs may appear as antimedian and median sets, respectively, for clients located at all vertices without repetitions, with the distance between them being arbitrary, see for instance [2]. It is possible that facilities which are undesirable for some clients may be desirable for some other clients. For example, assume the problem of locating a beer parlour in a human habitat area. Some of the inhabitants may consider it as a desirable facility where as some others may consider it as undesirable facility. One way to formulate such problem is to associate a sign with the clients indicating whether the facility is desirable or undesirable to the client. In this paper we are concentrating on methods to solve such problems. For this a more general concept called signed profiles is introduced and is formally defined in the next section. In Section 3, the equivalence of rational weight functions and signed profiles are established, and the relationship between the median and antimedian sets for signed profiles is obtained. In Section 4, various consensus strategies are formulated, amongst which Majority, Plurality and Scarcity strategy, and it is shown that all these consensus strategies are pairwise distinct for signed profiles, as it has already been known for the usual profiles. We show that, for signed profiles, the hypercubes are the only graphs on which Majority produces the median set for any signed profile in Section 5. Finally, for Scarcity, we study various classes of graphs, on which this strategy produces the antimedian set for any signed profile. 2 Preliminaries Let G = (V, E) be a finite, connected, simple graph with vertex set V and edge set E. The distance function of G is denoted by d, where d(u, v) is the length of a shortest u, v-path. We call a subset W of V connected if it induces a connected subgraph in G. The interval I(u, v) between two vertices u and v consists of all vertices on shortest u, v-paths, that is: A profile on G is a finite sequence n = (x1,x2,..., xk) of vertices of G. The length of n is the number k = |n|. Note that, n being a sequence, multiple occurrences are allowed. In this paper we extend the concept of profile: a signed profile is a profile where a sign from {+, -} is added to each element. We write the sign of element Xi as Si. Thus a signed profile is a sequence n = (s1x1, s2x2,..., skxk). We call xi an element of n and si its sign. Note that with this usage, a vertex occurring k times in a profile occurs as k different elements in a profile. For an element x of n we denote its sign also by sx. For computational reasons, we identify a sign s also with the number si = +1 or -1, and talk about +1 or -1 as a sign. Thus we can take the sum of signs. As we will see below, a signed profile with all signs being +1, plays the role of the usual profile without signs. We call such a profile a positive profile. If all signs are -1, then the profile is negative. Since all our profiles are signed, we call a signed profile just a profile, and omit the adjective 'signed', except in the statements of lemmas and theorems (to avoid confusion with similar lemmas and theorems in the literature). A profile obtained from n by changing each si by —si is denoted by —n. The size of a profile n is defined as So, for positive profiles we have ||n|| = |n|, and for negative profiles we have ||n|| = —|n|. For an edge uv in G, we denote by nuv the subprofile of n consisting of the elements of n strictly closer to u than to v, and by nU the subprofile of all elements at equal distance form u and v. Note that a profile, by definition, has a positive length. However, for sub-profiles we allow the empty subprofile. For instance, a graph is bipartite if and only if the subprofile nV is empty for any edge uv and any profile n. In the literature we find such concepts as remoteness, median and antimedian of positive profiles, for e.g., see, [14] and [20]. These are all very natural and the definitions are in accordance with our intuition. Because the definitions for signed profiles are basically the same, we use the same terminology here. The remoteness of a vertex v to a profile n is defined as I(u, v) = {x | d(u, x) + d(x, v) = d(u, v)}. k ||n|| =£ Si i=1 k i=1 A permutation of the elements in a profile does not change remoteness. Because we are only interested in the remoteness to profiles, we will consider two profiles as the same if they can be obtained from each other by permuting the elements. We write the concatenation of two profiles n and p as n p. Thus, for any edge uv, we can write n = nuvnUnvu. A vertex minimizing D(v, n) is called a median of the profile. The set of all medians of n is the median set of n and is denoted by M(n). A vertex maximizing D(v, n) is called an antimedian of the profile. The set of all antimedians of n is the antimedian set of n and is denoted by AM (n). The reader has to keep in mind that the effects of the signs might just be contra-intuitive. For instance, if n is a negative profile, then a median of n is an antimedian of the positive profile —n. A vertex x such that D(x, n) < D(y, n), for all neighbors y of x, is a local median of n. The set of all local medians is denoted by Mloc(n). If D(x, n) > D(y, n), for all neighbors y of x, then x is a local antimedian of n. The set of all local antimedians is denoted by AMloc(n). Let n = (sixi, s2x2,..., skxk) be a profile, then we have k k D(v, —n) ^^ —Sjd(xi, v) = — Sid(xi, v) = —D(v, n). i=i i=i From this observation we deduce that, by replacing a profile n by its opposite —n, the roles of (local) medians and (local) antimedians are exchanged. So we have M(n) = AM(—n), etcetera. We single out one fact that we need in the sequel. When the profile is of the form (—x, +x) the median set is equal to the antimedian set and is the entire vertex set of the graph. But this is not the only case when the median and antimedian set are equal. For example, consider a positive profile n on a hypercube containing each vertex once. In this case the remoteness is constant and hence the median and antimedian set are the same and coincide with the entire vertex set of the graph. It can also be noted that the situation is the same for —n. In general, for such positive and negative profiles on so called distance balanced graphs both the median sets and antimedian sets coincide. The case for such positive profiles on the class of distance balanced graphs is proved in [5]. The same situation holds for some special even profiles (both positive and negative) in some other class of graphs, see for instance [1]. Lemma 2.1. Let G be a connected graph and n a signed profile on G. Then, for any two adjacent vertices u, v in G, ||nuvII < ||nvu|| ifandonlyif D(u,n) > D(v,n). Proof. Since uv is an edge in G, we can ignore nu in the following computation. D(u,n) — D(v,n) = sxd(u, x) + sxd(u,x) — sxd(v,x) — ^^ sxd(v,x) ■£nuv xEnvu xEnuv x£nvu y: Sxd(u,x)+ Sxd(u,x) — Sx(d(u,x) + 1) — xEnuv xEnvu xGn. y: Sx(d(u,x) — i) x x£n„„ = KuH — ||nuv | From this the assertion follows immediately. □ 3 Remoteness with respect to arbitrary weight functions The concept of remoteness function and hence of medians and antimedians can also be studied with respect to weight functions defined on the vertex set of a graph. This was studied by Bandelt and Chepoi in [6] for non-negative weight functions in the case of medians. The equivalence of non-negative weight functions and positive profiles and hence the corresponding equivalence of the remoteness function and medians of non-negative weight functions and positive profiles are established in [4]. In this section, we establish that the same conclusion follows for arbitrary weight functions and signed profiles. A weight function on G is a mapping f from V to the set of real numbers. Note that we now allow negative weights. We say that f has a local minimum at x G V if f (x) < f (y), for every y adjacent to x. It has a local maximum if f (x) > f (y), for every y adjacent to x. The remoteness function with respect to the weight function f is the function Df from V to the set of real numbers defined as: Df (v) = D(v,f ) = £ d(v,x)f (x). xev Note that Df is a weight function on G as well. A local median of f is a vertex u such that Df has a local minimum at u. A local antimedian is a vertex at which Df attains a maximum. The set of all local medians of a weight function f is denoted by Mloc(f ). The set of all local antimedians is denoted by AMloc(f ). A median of f is a vertex u such that Df has a global minimum at u. Similarly, an antimedian of f is a vertex at which Df attains a maximum. The median set M (f ) of f is the set of all medians of f. The antimedian set AM( f ) of f is the set of all anti-medians of f. Let f be a weight function on a graph G and let -f be the weight function defined in the obvious way: its value at x is -f (x). Then clearly, we have D(v, f ) = —D(v, -f ), for any vertex v in G. In the sequel we make use of the following obvious facts. Remark 3.1. Let f be an arbitrary weight function defined on the vertex set of a graph G. Then replacing f with - f interchanges the roles of local maxima (minima) of f with local minima (maxima) of -f, and hence also interchanges the roles of both local and global medians (antimedians) of f with local and global antimedians (medians) of -f, respectively. Let n be a profile on G. Then the weight function associated with n is the function fn with fn(x) = J2 si, where the summation is taken over the occurrences of vertex x. If x does not occur in n, then we set f (x) = 0. The following lemma follows immediately from the definitions. Note that, for any integer-valued weight function f, there are infinitely many profiles having f as their associated weight function. Lemma 3.2. Let G be a connected graph, and let n be a signed profile with associated weight function fn. Then D(v,n) = D(v,fn), and hence M(fn) = M(n), and AM (fn ) = AM (n), and Mof ) = Mioc(n), and AMof ) = AMoc(n), for every v in V. Let f be a weight function on a connected graph G. For a positive real number t, we define tf to be the weight function with (tf )(x) = t x f (x). Then we have M(tf) = M(f ) and Mioc(tf ) = M(oc(f). Also we have AM(tf ) = AM(f ) and AMloc(tf) = AMloc(f ). Finally, Dtf has a strict local minimum (maximum) at a vertex u if and only if Df has a strict local minimum (maximum) at u. The following lemma is obvious. Lemma 3.3. Let g be rational weight function on a connected graph G. Then there is a signed profile n on G such that fn = tg for some positive integer t. In other words, antimedians (medians) of signed profiles are exactly antimedians (medians) of rational weight functions. The same holds for local antimedians (medians). Next we show that real-valued weight functions may be replaced by rational-valued weight functions, and thus by profiles, when one wants to compute antimedian (median) sets. We only present the proofs for the antimedian case. This is the one that we need in Sections 4 and 5. The case for the median sets is similar to that in [4], except that one has to take into account the signs. The next two Lemma's are the signed version of Lemma's 5 and 6 in [4]. The proofs are easy adaptations of those in [4]. Because they are short and prepare the way for Proposition 3.6, we include the proofs of the signed versions. Lemma 3.4. Let G be a connected graph, and let f bea weight function on G such that D f has a local maximum at vertex u, which is not a global maximum. Then there is a weight function g such that Dg has a strict local maximum at u, which is not a global maximum. Furthermore if f is rational, then g may also be taken as a rational function. Proof. First note that, for any two vertices x and y, we have d(x,y) < n = \V|. Let D(u, f ) = ei. Let Df have a global maximum at z, that is, D(z,f ) = e > ei. Let e2 = e - ei. Now define the function g as follows. g(v) = \ f(v) if v = u g(v) = t f (u) - f if v = u. Then D(u, g) = D(u, f ), because in these sums the values f (u) and g(u) of the functions at u are multiplied by d(u, u) = 0. For any vertex v adjacent to u, we have D(v, g) = D(v, f ) - < D(v, f ) < D(u, f ) = D(u, g). n So Dg has a strict local maximum at u. Furthermore, D(z,g) = D(z,f ) - d(u,z)> D(z,f ) - e2 = D(u, f ) = D(u,g). n So g has a strict local maximum at u that is not a global maximum. Also if f is rational, then e2 is rational. So g is also rational. Lemma 3.5. Let G be a connected graph with the property that, for each rational weight function g, every local maximum of Dg is also a global maximum. Then the same property holds for any real-valued weight function f on G. Proof. Assume that for some real-valued weight function f there is a local maximum for Df, at some vertex u that is not a global maximum. In view of the preceding lemma, we may assume that Df has a strict local maximum at u. Let Df have a global maximum at z, and let ei = min{D(u, f ) - D(x, f ) | x adjacent to u}, e2 = D(z, f ) - D(u, f ), _ min(ei, e 2 * n2 Now choose a rational weight function g such that g(v) < f (v) and f (v) - g(v) < e, for all v. Then, for any vertex x adjacent to u, we have D(u, g) > D(u, f ) - e x n2 > D(u, f ) - e1 > D(x, f ) > D(x, g). So u is a local maximum for Dg. Moreover, we have D(z, g) > D(z, f ) - e x n2 > D(z, f ) - e2 > D(u, f ) > D(u, g)* So u is not a global maximum for Dg, which is a contradiction. □ Graphs with connected median sets for non-negative weight functions were characterized in [6]. Using an analogous approach, we now are able to characterize graphs with connected antimedian and median sets for arbitrary weight functions. Before stating the result, we define basic concepts used in the following lines. A subgraph G of a graph H is an isometric subgraph if dG(u, v) _ dH(u, v) for all vertices u, v in G. We call a subset S of the vertex set of G a level set with respect to an integer A if S _ {x g V (G) : Df (x) > A}. Proposition 3.6. For a graph G and any arbitrary weight function defined on the vertex set of G the following conditions are equivalent (i) AMioc(f ) _ AM (f ) for all weight functions f; (ii) all level sets {x : Df (x) > A} induce isometric subgraphs; (iii) all antimedian sets AM(f ) induce isometric subgraphs; (iv) all antimedian sets AM(f) are connected. Proof. The implications (ii) ^ (iii), (iii) ^ (iv) are trivial. Next we prove (iv) ^ (i). Let f be a weight function. Assume to the contrary that there exists a local antimedian z of f that is not an antimedian. Let y be an antimedian. Amongst such pairs y, z, we may choose y and z such that d(y, z) is as small as possible. Our aim is to find two vertices u and v with d(u, v) _ 2 and a weight function f ' such that AM( f ') _ {u, v}. So f ' does not have a connected antimedian set. Consider the interval I(y, z). Because of the minimality of d(y, z), we have Df (y) > Df (x) for all x in I(y,z) distinct from y. Since z is a local antimedian, we have Df (z) > Df (x), for any neighbor x of z, in particular for any neighbor x of z in I(y, z). This implies that d(y, z) > 2. Hence, going from y to z within I(y, z), we will encounter two vertices u, v such that d(y, u) _ d(y, v) - 2, d(z, u) _ d(z, v) + 2, d(u, v) _ 2, with the properties that Df (u) > Df (x) and Df (v) > Df (x), for any common neighbor x of u and v. Note that these common neighbors of u and v are precisely the vertices in I(u, v) distinct from u and v. If there is any common neighbor x of u and v such that Df (v) _ Df (x), then we have Df (y) > Df (u) > Df (v). If Df (v) > Df (x) for all common neighbors of u and v, then we compare Df (u) and Df (v). If Df (u) > Df (v), then again we have Df (y) > Df (v). If Df (v) > Df (u), then we have Df (y) > Df (v) > Df (u). In this case we interchange the names of u and v. In all cases we end up with two vertices u and v at distance 2 with Df (y) > Df (u) > Df (v) > Df (x), for all common neighbors x of u and v, such that, additionally, Df (y) > Df (v) and Df (u) > Df (x), for all common neighbors x of u and v. We set p = Df(u)2Df (v). So p > 0, and Df (v) = Df (u) — 2p. We set p2 = Df (y ) — Df (v). Then p2 > p1 and p2 > 0. Note that for any x in V, we have Df (v) > Df (x) — M2. We construct the new weight function f ' from f as follows f (x) — (p + p) if x = v f (x) — (p2) if x = u f(x) otherwise. Straightforward computation now yields Df / (u) = Df (u) — 2(p + p) = Df (v) — 2p = Df, (v); and for any vertex x in I(u, v) distinct from u and v: Df i (x) = Df (x) — p — 2p < Df (u) — p — 2p2 = Df > (u) — p2 < Df > (u); and for any vertex x outside the interval (recall that p1 < p2): Df i (x) < Df (x) — 3p2 — P1 < Df (u) — 2p2 — P1 < Df i (u). Thus AM( f ') = {u, v}, and hence the antimedian set of f ' is not connected. This impossibility proves this implication. It remains to prove that (i) ^ (ii). Let AMloc(f ) = AM(f ) for all weight functions f. Assume to the contrary that the level set S = {x | Df (x) > A} corresponding to A is not isometric. Hence there exist two vertices u, v such that no shortest u, v-path lies completely inside S. Obviously, we can select u, v in S such that Df (x) < A for any x in I(u, v), distinct from u and v. Without loss of generality we may assume Df (u) < Df (v). Set Df (z) — Df (u) = p1, where z is an antimedian, and set e = min{Df (u) — Df (w) | w e I(u, v)}. Note that, since Df (u) > A > Df (w), for w in I(u,v), we have e > 0. Let P2 = d(U v). Define a weight function f ' such that {f (x) — P1 if x = u f (x) — (p1 + P2) if x = v f(x) otherwise. Straightforward computation now yields Dfi (v) = Df (v) — d(u,v)p1 > Df (u) — d(u,v)(p1 + P2) = Df i (u) and for any vertex w in I(u, v) distinct from u and v: Df i (w) < Df (w) — d(u, v)p1 < Df (u) — £2 — d(u, v)p1 = Df (u) — d(u, v)(p1 + P2) = Df i (u) and for any other vertex x: Df i (x) < Df (x) — (d(u, v) + < Df (z) — — d(u, v)^\ = Df (u) — d(u, v)^i = Df i (u). This implies that v is the unique antimedian of f ', while u is a local antimedian, which is not an antimedian vertex. This contradicts the assumption, by which the proof is complete. □ Above we established the equivalence of real-valued weight functions, rational-valued weight functions, and signed profiles with respect to medians etcetera. The next theorem is now an easy consequence of the previous results. Theorem 3.7. Let G be a connected graph. Then the following conditions are equivalent. (i) The antimedian set AM (f ) is connected, for all weight functions f on G. (ii) AM (f ) = AMioc(f ), for all weight functions f on G. (iii) The median set M (f ) is connected, for all weight functions f on G. (iv) M (f ) = Mloc(f ),for all weight functions f on G. (v) AM (f ) = AMloc(f ), for all rational weight functions f on G. (vi) AM (n) = AMloc(n), for all signed profiles n on G. (vii) M (f ) = Mloc(f ),for all rational weight functions f on G. (viii) M (n) = Mloc(n), for all signed profiles n on G. Proof. (i) up to (iv) are equivalent by Proposition 3.6, and Remark 3.1 . (ii) ^ (v) follows trivially. (v) ^ (ii) follows from Lemma 3.5. (v) ^ (vi): Let n be a signed profile on G. Now consider its associated weight function fn. By Lemma 3.2, we have D(v, fn) = D(v, n). Since Dfn cannot have any local maximum that is not a global maximum, Dn also cannot have any local maximum that is not a global maximum. (vi) ^ (v): Let g be any rational weight function on G. By Lemma 3.3, there is a positive integer t and a signed profile n such that fn = tg. By Lemma 3.2, Dfn = Dn, and, as observed above, D fn has a local maximum that is not a global maximum if and only if Dg have a local maximum that is not a global maximum. So Dg cannot have a local maximum that is not a global maximum. The equivalence of (vii) and (viii) with the other statements follows similarly. □ 4 Consensus Strategies If one wants to find the median set of a positive profile in a tree, then there exists a simple strategy formulated by Goldman [12] already in 1971. It reads as follows. When at vertex u, consider neighbor v of u. If there is a majority of the profile closer to v than to u, then move to v. In [20] this Majority Strategy was formulated for arbitrary graphs. There it was proved that the Majority Strategy produces the median set for any positive profile starting at any vertex if and only if the graph is a median graph, Theorem 4.1 below. For more details, we refer the reader to [20, 4]. A connected graph G is called a median graph, if every triple of vertices in G has a unique median. One of the main reasons underlying this result is that the structure of median sets is very nice in median graphs. In [4] four other related consensus strategies for positive profiles are studied. Antimedian sets are not so well-structured. So one cannot expect such deep results for signed profiles. But it is still possible to obtain some nice and unexpected results. Below we present a number of consensus strategies for signed profiles similar to the Majority Strategy from [18]. They are analogues of those in [4], but now formulated for signed profiles. In all the strategies below the input is a connected graph G, a profile n, and an initial vertex at which the strategy starts. There are two possibilities: one gets stuck at a vertex, or it is possible to visit vertices more than once. In the latter case the strategy could get into a loop, so the stopping rule must be more sophisticated here. In all cases, the output after stopping is the single vertex where one gets stuck or the set of vertices visited at least twice. Steps 1, 3 and 4(i) below are the same for all strategies, so we list these only in the first instance. In all other instances we only list Step 2, describing when one moves to a neighbor, and Step 4(ii), the stopping rule when one does not get stuck. Majority Strategy 1. Start at the initial vertex. 2. [MoveMS] If we are in u and v is a neighbor of u with ||nvu|| > 2 ||n||, then we move to u. 3. We move only to a vertex already visited if there is no alternative. 4. We stop when (i) we are stuck at a vertex u or (ii) [TwiceMS] we have visited vertices at least twice, and for each vertex u visited at least twice and each neighbor v of u, either ||nvu|| < 2 ||n|| or v is also visited at least twice. Before presenting the other strategies we quote the main theorem from [20]. This theorem has been the motivation for studying such strategies on graphs. It also shows the special role of median graphs within the Class of All Graphs. Due to the structure theory of median graphs, the equivalence of (ii) and (iii) on median graphs in the theorem is not surprising. But otherwise it would not have been something one would expect at first sight. Theorem 4.1. [Majority Theorem] Let G be a graph. Then the following conditions are equivalent. (i) G is a median graph. (ii) The Majority Strategy produces the median set M(n) from any initial vertex, for each positive profile n on G. (iii) The Majority Strategy produces the same set from any initial vertex, for each positive profile on G. In the majority strategy one moves towards majority. A slightly different point of view is to move away from minority. This seems to be the same, but it is not, as we will see below. This latter strategy is known as the Condorcet Strategy. Condorcet Strategy 2. [MoveCS] If we are in u and v is a neighbor of u with ||nuv || < 2 ||n||, then we move to v. 4. (ii) [TwiceCS] we have visited vertices at least twice, and for each vertex u visited at least twice and each neighbor v of u, either ||nuv || > 2 ||n|| or v is also visited at least twice. In non-bipartite graphs the subprofile nU of n, for an edge uv, is not always empty. From the viewpoint of voting, one might say that the elements of nU abstain from voting when the choice is between u and v. So these may be ignored when the question is whether to move from u to v. This is the idea behind the Plurality Strategy. Note that on bipartite graphs Majority and Plurality coincide. Plurality Strategy 2. [MovePS] If we are in u and v is a neighbor of u with ||nvu|| > ||nuv ||, then we move to v. 4. (ii) [TwicePS] we have visited vertices at least twice, and for each vertex u visited at least twice and each neighbor v of u, either ||nvu || < ||nuv || or v is also visited at least twice. The next two strategies were introduced to find a (local) minimum based on a heuristic function in a search graph. They are also known as Hill Climbing and Steepest Ascent Hill Climbing, respectively. Ascent Strategy 2. [MoveAS] If we are in u and v is a neighbor of u with D(v, n) < D(u, n), then we move to v. 4. (ii) [TwiceAS] we have visited vertices at least twice, and for each vertex u visited at least twice and each neighbor v of u, either D(v, n) > D(u, n) or v is also visited at least twice. Steepest Ascent Strategy 2. [MoveSAS] If we are in u and v is a neighbor of u with D(v, n) < D(u, n), and D(v, n) is minimum among all neighbors of u, then we move to v. 4. (ii) [TwiceSAS] = [TwiceAS]. The next simple Lemma is an analogue of Lemma 1 in [4] for signed profiles with the same conclusion. Note that the Plurality and Ascent strategy produce the same output for signed profiles on any connected graph. On bipartite graphs both coincide with Majority. Lemma 4.2. Let G be a connected graph and n a signed profile on G. Plurality Strategy makes a move from vertex v to vertex u if and only if D(u, n) < D(v, n). Proof. The assertion follows immediately from the following computation: D(v, n) — D(u, n) = sxd(v,x) + > sxd(v,x) — > sxd(u,x) ^^ sxd(v, x) + sxd(v, x) — ^^ sxd(u,x) — ^^ sxd(u,x) xGnvu xGnuv xGnvu xGnuv = £ Sxd(v,x)+ £ Sxd(v, x) — £ sx(d(v,x) + 1) — xGnvu xGnuv xGnvu £ Sx(d(v, x) — 1) xG^uv = Ikuvll - Ikvull. □ The various strategies are quite similar. But on general graphs they are all different. We present some examples to show this. The first example shows that Plurality, Condorcet and Majority strategies are pairwise distinct. Consider the profile n = (+a, +b, -c, +d, +e, +f) on the graph shown in Figure 1. We have ||n„v|| = 1, |K„|| = 0, IKJ = IKbH = |Kd|| = ||nve| = 2 ||nvc|| = 4, ||n0v|| = ||n6v|| = ||ndv|| = || = 1, ||nCv|| = -1, ||n„o|| = ||n„b|| = ||n„d|| = ||n„e|| = ||n„/1| = 3, ||n„c|| = 5, ||no„|| = ||n6«y = ||nd«y = ||ne«y = ||n/j =l, ||nc«y = -1. a b c d e f Figure 1: Consensus strategies differing on a graph Apply all the strategies starting at u. Using Majority we may not move to any of its neighbors, so we are stuck at u. Thus the outcome of Majority is {u}. Note that we have ||nux|| < 2 for any neighbor x of u other than c. So, if we use Condorcet, then we can move to any of its neighbors except c. Note also that from a, b, d and e a move to either u or v is allowed, but from v we can move only to u. Thus using Condorcet we may move along u, a, b, d, e, v. Hence the output of the Condorcet Strategy is {u, a, b, d, e, v}. When we use Plurality, then we can move only to v and we get stuck at v. Hence the output of Plurality is {v}. Ascent and Steepest Ascent strategies also produce the output {v}. It is shown in [4] that Steepest Ascent is essentially different from the other strategies for positive profiles. Note that the other strategies might make a move from u as soon as they find a neighbor v of u that satisfies the condition for a move, while Steepest Ascent has to check all neighbors of u before it can make a move. For a comparison of efficiencies of these strategies, see [3]. The following example shows that the first four strategies might not even find the median vertex, even if the graph is bipartite. Consider the complete bipartite graph K2,5 with vertices a, b and 1, 2, 3,4,5, where two vertices are adjacent if and only if one is a 'letter' and the other is a 'numeral'. Now take the profile n = (+b, +1, +1, +1, +2, +2, +2, +3, +3, +3, +4, -5). Then we have D(a, n) = 11, D(b, n) = 9, D(4, n) = 21, D(5, n) = 17 and D(i, n) = 13, for i = 1,2, 3. Take 1 as initial vertex and assume that we check its neighbors in alphabetical order. Then Majority, Condorcet, Plurality and Ascent strategies move to a and get stuck there, whereas Steepest Ascent moves to b and thus finds the median vertex of n. It was already shown in [4] that Plurality produces the median set for any positive profile if and only if all median sets in the graph are connected. Hence Plurality produces the antimedian set for negative profile in such graphs. Moreover the antimedian sets are connected for all negative profiles in such graphs. In the case of finding antimedian sets one would want to have the "converse" of the above strategies, that is, apply the strategy on —n instead of n. Because we are working with signed profiles these are strategies in their own right. We list them below, with their appropriate names. Minority Strategy Minority applied to n is identical with Majority applied to —n. Scarcity Strategy Scarcity applied to n is identical with Plurality applied to —n. Descent Strategy Descent applied to n is identical with Ascent applied to —n. Steepest Descent Strategy Steepest Descent applied to n is identical with Steepest Ascent applied to —n. It is interesting to note that Scarcity produces the antimedian set in hypercubes. Recall that the n-dimensional hypercube Qn, the n-cube for short, has the 0,1-vectors of length n as its vertices, two vertices being adjacent if the corresponding vectors differ in exactly one coordinate. Take any i with 1 < i < n. Let Q0n i be the (n — 1)-dimensional subcube consisting of the vertices with a 0 as i-th coordinate, and let Qjl,i be the complementary subcube consisting of the vertices with a 1 as i-th coordinate. For a profile n, let n? be the subprofile of n in Qn ^ and let nj to the subprofile of n in Qn i. Let W be the set of vertices in Qn, for which n has a signed minority in each coordinate. That is, x lies in W if and only if x has a 0 in the i-th coordinate when ||nj|| > ||n0||, and a 1 when ||n0|| > ||nj||, and a 0 or 1 when ||n0|| = Hn/H. This is precisely the antimedian set of n. It is also a subcube of dimension d, where d is the number of coordinates, for which 11 n? 11 = 11 n j11. Proposition 4.3. Scarcity strategy produces the antimedian set on a hypercube for any signed profile. 10 Proof. Take any i with 1 < i < n. Take any vertex u in Qn, i, and let v be its neighbor in Qn , i. Then we have nuv = n0 and nvu = nj. So, if ||n0|| > 11 n11, then we move from Qn i to Qn i. And if ||n0|| > ||nj ||, then we never move back to Q0n i. So Scarcity moves to the set of vertices W in Qn, for which n has a signed minority in each coordinate. This is precisely the antimedian set of n. □ In general Scarcity will not always produce an antimedian. For example when we use Scarcity moves in a tree we will always stuck at leaf nodes, as we can see in the following lines. Consider a tree T with at least three leaves, and a positive profile n of length k on T. Take any leaf v that occurs less than j k times in n, and let u be the neighbor of v in T. Note that d(v, x) = d(u, x) + 1 for any x = v in T. Obviously we will move to v from u, but we will never move back to v using Scarcity. But v need not be the antimedian of n. If the profile is "close" to u, then obviously antimedian will be at some other leaves far away. Next we prove an analogue of the main Theorem (Theorem 8 in [4]) for positive profiles in the case of signed profiles. Theorem 4.4. The following are equivalent for a connected graph G. (i) The Scarcity Strategy produces AM(n) from any initial vertex, for all signed profiles n on G. (ii) AM (n) is connected, for all signed profiles n on G. (iii) AM (n) = AMloc(n),for all signed profiles n on G. (iv) Descent Strategy produces AM (n) from any initial vertex, for all signed profiles n on G. (v) Steepest Descent Strategy produces AM (n) from any initial vertex, for all signed profiles n on G. (vi) Scarcity, Descent, and Steepest Descent Strategy each produce the same set from any initial vertex, for all signed profiles. Proof. (i) ^ (ii): Suppose the antimedian set is not connected for some profile n. Then let u and v be two vertices in different components of AM(n). Now, if Scarcity starts at u, then it cannot reach vertex v, because a move from an antimedian vertex to a non-antimedian vertex is not possible by Lemma 4.2. So the set computed by Scarcity will not include u, which is a contradiction. (ii) ^ (iii): This follows from Theorem 3.7. (iii) ^ (iv): Starting at any vertex, Descent always finds a local maximum. Since this local maximum is also global, it follows that Descent always reaches an antimedian, and since the antimedian set is connected, Descent finds all antimedian vertices. (iv) ^ (i): Assume that Descent finds the antimedian set. This means that Descent will move to an antimedian starting from any vertex and finds all the other antimedians. The same moves will be made by Scarcity, by Lemma 4.2. Hence Scarcity will compute the antimedian set correctly. (iii) ^ (v) follows similarly as (iii) ^ (iv). (v) ^ (ii) follows from the fact that Steepest Descent finds a local maximum and does move from antimedian to antimedian but does not move from an antimedian to a non-antimedian. (i) ^ (vi) is obvious. (vi) ^ (i) follows from the fact that, starting from an antimedian, Scarcity can produce only a set of antimedians which includes the initial vertex. So starting from any vertex it produces the same set if and only if the produced set is actually AM(n). The same argument works for Descent and Steepest Descent. □ 5 The Majority Strategy for Signed Profiles In this section a characterization for hypercubes is obtained as the graphs for which Majority always produces the median set for any signed profile. Before stating the result, we need a few facts from the theory of median graphs as developed in [18]. A median graph is bipartite, and does not contain K2}3 as induced subgraph. This implies that any two vertices at distance 2 have either one or two common neighbors. It is proved in [18] that a graph G is a hypercube if and only if it is a median graph in which any two vertices at distance 2 have exactly two common neighbors. For any vertex w in a graph G, we write Ni(w) = {x | d(x,w) = i}, and N>i(w) = \Jj>i Nj(w). Proposition 5.1. Let AM be the antimedian function on a median graph G. Then AM (n) is connected for every signed profile n if and only if G is a hypercube. Proof. If G is a hypercube, then Proposition 4.3 gives us the required result. Conversely, let G be a median graph for which the antimedian set is connected, for any signed profile. Let u and v be two vertices at distance 2, and let w be a common neighbor of u and v. Due to the above mentioned characterization of hypercubes in [18] we have to prove that there exists a unique common neighbor of u and v different from w. Consider the profile n = (+w) of length 1. Note that, for any x in Nj(w), we have Df (x) = j. So N>0(w) = V - w is a level set with respect to n. Due to Proposition3.6 any level set of n induces an isometric subgraph. So, within V - w, the vertices u and v have distance 2 as well, that is, there is a common neighbor z in V - x. Since a median graph is bipartite and does not contain K2 3, this neighbor is unique. □ The next theorem is an analogue of the majority theorem for signed profiles which turns out to be a new characterization of hypercubes. Theorem 5.2. A graph G is a hypercube if and only if the Majority Strategy, starting from any initial vertex, produces the median set for any signed profile on G. Proof. If G is a hypercube, then, by Proposition 4.3, Scarcity produces the antimedian set for any signed profile. So, since the hypercube is bipartite, Minority produces the antimedian set for any signed profile, whence Majority produces the median set for any signed profile. Conversely, assume that Majority produces the median set for any signed profile. Then it also produces the median set for any positive profile. So, by Theorem 4.1, the graph is a median graph. Hence, by Proposition 5.1, the graph is a hypercube. □ 6 Graphs for which Scarcity produces the Antimedian Set for any Signed Profile In this section we discuss some graph classes for which Scarcity always produces the anti-median set for any signed profile. First we consider the Hamming graphs. Let ki,... ,kn be positive integers, and let V be the Cartesian product nn=i{0,i,...,ki -1}. The Hamming graph Hkli...jkn is the graph with vertex set V, in which two vertices are joined by an edge if and only if the corresponding vectors differ in exactly one coordinate. The properties for Hamming graphs needed here probably all belong now to folklore, but could also be found in [18, 19], where they were characterized for the first time. The set of vertices in H = Hkli...jkn having a in the i-th position of the corresponding vector is denoted as Hj^ kni, or simply as Hf. For a profile n, we denote its subprofile contained in Hf by nf. Let n be a profile on H = Hkli...jkn. Fix a position i, for which ki > 2, and let a and b be distinct elements in {0,... ,ki - 1}. Let u be a vertex in Hf, and let v be its neighbor in Hb. Then nuv = nf and nvu = n\. Note that, if u is in AM(n), then we have ||nf|| < ||nf ||. This holds for every b in {0,... ,ki - 1} distinct from a. In this case we say that there is a signed minority at a in position i. Clearly, an antimedian vertex in H is a vertex with a signed minority in each coordinate. Let m. be the number of elements in {0, * * *,k. - 1} with a signed minority, for i _ 1, * * * ,n. Then the antimedian set of n induces a subgraph isomorphic to Hmi}.,mn. Obviously, any antimedian set is connected. Proposition 6.1. Starting form any vertex Scarcity Strategy produces the antimedian set on a Hamming graph for any signed profile. Proof. Let n be a (signed) profile on the Hamming graph H _ Hkl..,kn. Take any i with 1 < i < n and any vertex u in H", and let v be its neighbor in Hb with b _ a. If ||n"|| > ||nb||, then we move from H" to Hb. And if ||n"|| > ||nb||, then we never move back to H". So Scarcity moves to the set of vertices W in H for which n has a signed minority in each coordinate. This is precisely the antimedian set of n. □ Graphs which admit a scale-A, (A > 2) embedding into a hypercube is called an l1 graph. The Johnson graphs and half cubes are important classes of l1 graphs which occur as hosts for isometric embeddings of graphs, [8]. Next we consider the Johnson graphs followed by half cubes. The Johnson graph Jn}k has as vertices the k-element subsets of {1, 2, * * *, n}, and two vertices are adjacent if and only if their intersection has size k - 1. In other words the vertices 'differ' in exactly one element. Some special Johnson graphs are: Jn<1 is the complete graph on n vertices, Jn<2 is the n-triangular graph, and Jnß is n-tetrahedral graph. Since each vertex u in Jn k. Clearly, if m < k - r, we get a vertex in G by changing a minimum number of coordinates, say p, from the vertex in M(n, Qn) having 1s in positions 1, * * * ,r. Similarly, when m > k, we get a vertex in G with a minimum number of changes, by selecting the vertex with 0's in positions 1,..., r. Since we are looking for a median set in G, we select the positions, in such a way that the change in remoteness is minimum. Thus we select p coordinate positions with smaller signed majority values. If the signed majority values are distinct we get a single vertex in G. Otherwise we make a selection among, say p' positions. In this case the subgraph induced by the vertices of G thus obtained will be isomorphic to Jp>. Since the remoteness is same for all vertices in the median set we get the same result independent of the vertex selected. Hence we get a subgraph that is a Johnson graph as the median set. With similar arguments, by taking the signed minority values at coordinate positions, we can prove that antimedian sets also induce some Johnson graph. This completes the proof. □ From the above theorem we have the following corollary. Corollary 6.3. Let G be a Johnson graph. Then M(n) and AM(n) are connected, for any signed profile n in G. From the above Corollary and Theorem 4.4 we have: Corollary 6.4. Starting from any vertex on a Johnson graph Scarcity strategy produces the antimedian set for any signed profile. Next, we consider halfcubes. The vertex set of a halfcube is the subset of the vertices of the hypercube Qn with an even (respectively, odd) number of ones in their vector representation. Two vertices are adjacent when they differ in exactly two positions, see [8]. Halfcubes also admit a scale-2 embedding into the corresponding hypercube. Theorem 6.5. Let G be a halfcube, then M (n, G) and AM (n, G) are connected for any signed profile n in G. Proof. Let Qn be the hypercube of dimension n in which G is scale-2 embedded. Let n be an arbitrary profile in G and ||n|| = k. Note that by applying the Majority rule for the given profile n of the halfcube embedded into hypercube Qn (looking as the vertices of a hypercube), we get the median of n in Qn which will be a sub-hypercube, say Qr. We analyze the property of M (n, G) by considering the following two cases separately. Case 1. M(n, Qn) is a hypercube Qr of dimension at least one. Clearly Qr has half vertices in the corresponding halfcube - call this set X. Set X forms a halfcube in G, hence X is connected. Since the graph G is scale-2 embedded the remoteness in G is obtained by dividing the corresponding remoteness in Qn by 2, we get M (n, G) = X, as we follow the signed Majority rule on n. Case 2. M(n, Qn) in Qn contains exactly one vertex say x. If x belongs to G, then clearly M (n, G) = {x} as the case may be and hence we are done. So assume that x is not in G. Note that x = (x1,..., xd) can be obtained from the signed Majority rule among coordinates of the profile n. Let m,, 1 < m, < n be the signed majority at each position. Let m = min{m1,..., mn}. Clearly if for any vertex y obtained by changing any single ith coordinate of x, the remoteness changes by 2m, - k, where ||n|| = k. This change in remoteness is minimum for coordinates having signed Majority value m. Hence M (n, G) is precisely the set of vertices obtained from G by changing any coordinate of x, having minimum signed Majority mž. These vertices are all adjacent to x, and hence forms a clique in G. Thus M (n, G) is connected for any signed profile. With similar arguments and by taking m as maximum(m1,..., mn), where each mž is signed minority, we can prove that AM (n, G) is also connected for any profile, which completes the proof. □ From the proof of the above theorem, we have the following corollary. Corollary 6.6. Let G be a halfcube, then M (n, G) and AM (n, G) induce a halfcube in G or a clique, for any profile n in G. From Theorem 6.5 and Theorem 4.4 we have: Corollary 6.7. Starting from any arbitrary vertex in a halfcube Scarcity Strategy always produce antimedian set for any signed profile 7 Concluding remarks In this paper, we have proved that the classes of graphs in which the consensus strategies Scarcity, Descent and Steepest Descent will always produce the antimedians for any arbitrary signed profile is precisely the class of graphs with connected antimedians. This class of graphs is characterized in terms of (local) medians and (local) antimedians of (rational) weight functions. Also, we proved that, among the median graphs, the hypercubes are precisely the graphs with connected antimedians for an arbitrary signed profile. Moreover, we presented some classes on which Scarcity produces the antimedian set for any signed profile. An intriguing question remains: Which classes of graphs have connected antimedians for arbitrary signed profiles? References [1] K. Balakrishnan, B. Bresar, M. Changat, S. Klavzar, M. Kovse and A. R. Subhamathi, On the remoteness function in median graphs, Discrete Appl. Math. 157 (2009), 3679-3688. [2] K. Balakrishnan, B. Bresar, M. Changat, S. Klavzar, M. Kovse and A. R. Subhamathi, Simultaneous embeddings of graphs as median and antimedian subgraphs, Networks 56 (2010), 90-94. [3] K. Balakrishnan, M. Changat and H. M. Mulder, Median computation in graphs using consensus strategies, Report EI 2006, Econometrisch Instituut, Erasmus Universiteit, 2006, Rotterdam. [4] K. Balakrishnan, M. Changat and H. M. Mulder, The plurality strategy on graphs, Australsian J. Combin. 46 (2010), 191-202. [5] K. Balakrishnan, M. Changat, I. Peterin, S. Spacapan, P. Sparl and A. R. Subhamathi, Strongly distance-balanced graphs and graph products, Eur. J. Combin. 30 (2009), 1048-1053. [6] H. J. Bandelt and V. Chepoi, Graphs with connected medians, SIAM J. Discr. Math. 15 (2002), 268-282. [7] J. P. Barthelemy and B. Monjardet, The median procedure in cluster analysis and social choice theory, Math. Social Sci. 1 (1981), 235-268. [8] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989. [9] P. Cappanera, A survey on obnoxious facility location problems, Technical Report TR-99-11, University of Pisa, 1999. [10] P. Cappanera, G. Gallo and F. Maffioli, Discrete facility location and routing of obnoxious activities, Discrete Appl. Math. 133 (2003), 3-28. [11] R. L. Church and R. S. Garfinkel, Locating an obnoxious facility on a network, Transportation science 12 (1978), 107-118. [12] A. J. Goldman, Optimal center location in simple networks, Transportation Science 5 (1971), 212-221. [13] S. KlavZar and H. M. Mulder, Median graphs: characterizations, location theory and related structures, J. Combin. Math. Combin. Comput. 30 (1999), 103-127. [14] F. R. McMorris, H. M. Mulder and F. S. Roberts, The median procdure on median graphs, Discrete Appl. Math. 84 (1998), 165-181. [15] F. R. McMorris, H. M. Mulder and R. V. Vohra, Axiomatic characterization of location functions, in: H. Kaul and H. M. Mulder (eds.), Advances in interdisciplinary discrete applied math-ematcis, Interdisplinary Mathematical Sciences, Vol. 11, World Scientific, Singapore, 2010, pp. 71-91. [16] F. R. McMorris and R. C. Powers, The median procedure in a formal theory of consensus, Siam J. Discrete Math 8 (1995), 507-516. [17] P. B. Mirchandani and R.L. Francis, Discrete Location Theory, Wiley-Interscience, New York, 1990. [18] H. M. Mulder, The Interval Function of a Graph, Mathematical Centre Tracts 132, Mathematisch Centrum, Amsterdam, 1980. [19] H.M. Mulder, Interval-regular graphs, Discrete Math. 41 (1982), 253-269. [20] H. M. Mulder, The Majority Strategy on graphs, Discrete Appl. Math. 80 (1997), 97-105. [21] H.M. Mulder, Median graphs, A structure theory, in: H. Kaul and H.M. Mulder (eds.), Advances in interdisciplinary discrete applied mathematcis, Interdisplinary Mathematical Sciences, Vol. 11, World Scientific, Singapore, 2010, pp. 93-125. [22] R. C. Powers, Consensus centered at majority rule, in: H. Kaul and H.M. Mulder (eds.), Advances in interdisciplinary discrete applied mathematcis, Interdisplinary Mathematical Sciences, Vol. 11, World Scientific, Singapore, 2010, pp. 149-166. [23] R. C. Tansel, R. I. Francis and T. J. Lowe, Location on networks: A survey I, Management Sci. 29 (1983), 482-497. [24] R. C. Tansel, R. I. Francis and T. J. Lowe, Location on networks: A survey II, Management Sci. 29 (1983), 498-511. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 147-154 Quasi m-Cayley circulants Ademir HujduroviC * University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia University of Primorska, IAM, Muzejski trg 2, 6000 Koper, Slovenia Received 30 October 2011, accepted 4 February 2012, published online 15 June 2012 Abstract A graph r is called a quasi m-Cayley graph on a group G if there exists a vertex to G V(r) and a subgroup G of the vertex stabilizer Aut(r)TO of the vertex to in the full automorphism group Aut(r) of r, such that G acts semiregularly on V(r)\{ro} with m orbits. If the vertex to is adjacent to only one orbit of G on V(r)\{ro}, then r is called a strongly quasi m-Cayley graph on G. In this paper complete classifications of quasi 2-Cayley, quasi 3-Cayley and strongly quasi 4-Cayley connected circulants are given. Keywords: Arc-transitive, circulant, quasi m-Cayley graph. Math. Subj. Class.: 05C15, 05C10 1 Introduction Throughout this paper graphs are assumed to be finite, simple, connected and undirected, and groups are finite. Given a graph r we let V(r), E(r), A(r) and Aut(r) be the set of its vertices, edges, arcs and the automorphism group of r, respectively. A graph r is said to be vertex-transitive, edge-transitive, and arc-transitive if its automorphism group acts transitively on V (r), E (r) and A(r), respectively. Let G be a finite group with identity element 1, and let S c G\{1} be such that S-1 = S. We define the Cayley graph Cay(G, S) on the group G with respect to the connection set S to be the graph with vertex set G, in which two vertices x, y G G are adjacent if and only if x-1y G S. A circulant of order n is a Cayley graph on a cyclic group of order n. In this paper we consider quasi-semiregular actions on graphs, a natural generalization of semiregular actions on graphs, which have been an active topic of research in the last decades (see, for example, [1,2, 3,4, 5, 8,9,11]). Following [7] we say that a group G acts * Supported in part by "Agencija za raziskovalno dejavnost Republike Slovenije", research program P1-0285 and "Mladi raziskovalec" research program. E-mail address: ademir.hujdurovic@upr.si (Ademir Hujdurovic) quasi-semiregularly on a set X if there exists an element to in X such that G fixes to, and the stabilizer Gx of any element x e X\{to} is trivial. The element to is called the point at infinity. A graph r is called quasi m-Cayley on G if the group G acts quasi-semiregularly on V(r) with m orbits on V(T)\{to}. If G is cyclic and m = 1 (respectively, m = 2, m = 3 and m = 4) then r is said to be quasi circulant (respectively, quasi bicirculant, quasi tricirculant and quasi tetracirculant). In addition, if the point at infinity to is adjacent with only one orbit of then we say that r is a strongly quasi m-Cayley graph on G. Quasi m-Cayley graphs were first defined in 2011 by Kutnar, Malnic, Martinez and Marušic [7], who showed which strongly quasi m-Cayley graphs are strongly regular graphs. In this paper, we consider which circulants are also quasi m-Cayley graphs. Our main results are stated in the following three theorems. Theorem 1.1. Let r be a quasi 2-Cayley graph of order n which is also a connected circulant. Then either r is isomorphic to the complete graph Kn, or n = 1 (mod 4) is a prime and r is isomorphic to the Paley graph P(n). Moreover, r is a quasi bicirculant. Theorem 1.2. Let r be a connected circulant. Then r is also a quasi 3-Cayley graph if and only if either r = Kn, or replacing r with its complement if necessary, r = Cay( Zn, S), where S is the set of all non-zero cubes in Zn, and n is a prime such that n = 1 (mod 3). Moreover, r is a quasi tricirculant. Theorem 1.3. Let r be a connected circulant. Then r is a strongly quasi 4-Cayley graph on a group G if and only if r = C9 or r = Cay(Zn, S), where S is the set of all fourth powers in Zn\{0}, and n is a prime such that n = 1 (mod 4). Moreover, r is a quasi tetracirculant. The paper is organized as follows. In Section 2 we recall the classification of connected arc-transitive circulants. In Section 3 we prove Theorem 1.1 and in Section 4 we prove Theorems 1.2 and 1.3. 2 Arc-transitive circulants We begin this section with the following lemma: Lemma 2.1. Let r be a connected vertex-transitive strongly quasi m-Cayley graph. Then r is arc-transitive. Proof. Since r is vertex transitive, it is sufficient to prove that there exists a vertex v such that the stabilizer Aut(r)v acts transitively on the neighborhood of v. It is obvious that if we choose the point at infinity for v, this condition is satisfied. □ The previous lemma implies that we can somehow restrict our study to the connected arc-transitive circulants, therefore it is important to understand the structure of such graphs. To state the classification of connected arc-transitive circulants, which has been obtained independently by Koväcs [6] and Li [10], we need to recall certain graph products and the concept of normal Cayley graphs. The wreath (lexicographic) product S[r] of a graph r by a graph S is the graph with vertex set V (S) x V (r) such that {(wi,w2), (vi,v2)} is an edge if and only if either {u1,v1} e E(S), or u1 = v1 and {u2, v2} e E(r). For a positive integer b and a graph E, denote by bE the graph consisting of b vertex-disjoint copies of the graph E. The graph E [Kb] — bE is called the deleted wreath (deleted lexicographic) product of E and Kb, where Kb = bK i. Let r = Cay(G, S) be a Cayley graph on a group G. Denote by Aut(G, S) the set of all automorphisms of G which fix S setwise, that is, Aut(G,S) = {a G Aut(G)|SCT = S}. It is easy to check that Aut(G, S) is a subgroup of Aut(r) and that it is contained in the stabilizer of the identity element 1 G G. It follows from the definition of Cayley graph that the left regular representation GL of G induces a regular subgroup of Aut(r). Following Xu [12], r = Cay(G, S) is called a normal Cayley graph if GL is normal in Aut(r), that is, if Aut(G, S) coincides with the vertex stabilizer 1 G G. Moreover, if r is a normal Cayley graph, then Aut(r) = gl x Aut(G, S). Proposition 2.1. [6, 10] Let r be a connected arc-transitive circulant of order n. Then one of the following holds: (i) r = K„; (ii) r = E[Kd], where n = md, m, d > of order m; (iii) r = E[Kd] — dE, where n = md, arc-transitive circulant of order m; (iv) r is a normal circulant. In Section 3 and 4 two lemmas (that show that arc-transitive circulants described in Proposition 2.1(ii) and (iii) are not strongly quasi k-Cayley graphs) will be needed. Lemma 2.2. Let r be an arc-transitive circulant, described in Proposition 2.1(ii). Then r is not a strongly quasi k-Cayley graph for any k G N. Proof. We have r = E[Kd], where n = md, m,d> 1 and E is a connected arc-transitive circulant of order m. Suppose that r is a strongly quasi k-Cayley graph on a group G. Then val(r) = (n — 1)/k = (md — 1)/k. On the other hand, since r = E[Kd], we have val (r) = val(E) • d. These two facts combined together imply that d(m — k • v al (E)) = 1, and so d = 1, a contradiction. □ Lemma 2.3. Let r be an arc-transitive circulant, described in Proposition 2.1(iii). Then r is not a strongly quasi k-Cayley graph for any k G N. Proof. We have r = E[Kd] — dE, where n = md, d > 3, gcd(d, m) = 1, and E is an arc-transitive circulant of order m. Suppose that r is also a strongly quasi k-Cayley graph on a group G. By [10, Theorem 1.1] the m copies of the graph Kd form an imprimitivity block system B for Aut(r). Clearly the block B G B containing the point at infinity, that is, the trivial orbit of G, is fixed by G. This implies that | G| divides d — 1 . On the other hand, since the valency of r is |G|, we have |G| > d — 1. Combining these results we obtain |G| = d — 1. Thus, connectedness of r implies that m = 2. However, then there are 2d — 1 vertices in r different from the point at infinity, and they cannot be divided into k orbits of size d — 1 for any natural number k. Therefore, there are no strongly quasi k-Cayley graphs amongst the graphs from Proposition 2.1(iii) for any natural number k > 1. □ 1 and E is a connected arc-transitive circulant d > 3, gcd(d, m) = 1 and E is a connected Lemma 2.4. Let r be an arc-transitive circulant, described in Proposition 2.1(iv). If r is also a strongly quasi m-Cayley graph on a group G, then the order of r has at most m +1 divisors. Proof. Let r = Cay(Zn, S) be a normal circulant. Let A = Aut(T). Since r is a normal Cayley graph, A = Zn x Aut(Zn, S). We may, without loss of generality, assume that the point at infinity corresponds to the vertex 0 G Zn, and so G < Aut(Zn, S) < Aut(Zn) = Zn. Therefore, G < Zn. Since G has m orbits on Zn\{0}, then Aut(Zn) has at most m orbits on Zn\{0}, and at most m +1 orbits on Zn. Elements in the same orbit of Aut(Zn) are clearly of the same order in Zn. There exist an element in Zn of order d, if and only if d divides n. Therefore the number of divisors of n, denoted by t(n), is not greater than m + 1, i.e. t (n) < m +1. □ 3 Quasi 2-Cayley graphs In this section the connected circulants are considered. In particular, connected circu-lants that are also quasi 2-Cayley graphs are classified (see Theorem 1.1). If a graph r of order n is a quasi 2-Cayley graph on a group G, which is not a strongly quasi 2-Cayley graph, then it is isomorphic to the complete graph Kn. Namely, in such a graph, the point at infinity to is adjacent to both nontrivial orbits of G, and thus it is adjacent to all the vertices different from to. Consequently, we can conclude that r has valency | V(r) | - 1, and so r is a complete graph. In order to classify all connected circulants that are also quasi 2-Cayley graphs it therefore suffices to characterize strongly quasi 2-Cayley graphs that are also connected circulants, we do this in Theorem 3.1. Theorem 3.1. Let r be a connected circulant. Then r is also a strongly quasi 2-Cayley graph if and only if r is isomorphic to the Paley graph P (p), where p is a prime such that p = 1 (mod 4). Moreover, r is a quasi bicirculant. Proof. Let r be the Paley graph P (p), where p is a prime, such that p = 1 (mod 4). It is well known that the Paley graphs are connected arc-transitive circulants, and, as was observed in [7], they are also strongly quasi 2-Cayley graphs. Conversely, let r be a connected circulant Cay(Zn, S) of order n not isomorphic to the complete graph Kn, which is also a strongly quasi 2-Cayley graph on a group G. Then | G| = (n —1)/2 and r is of valency (n —1)/2. Lemma 2.1 tells us that r is an arc-transitive graph, and moreover Proposition 2.1, Lemma 2.2 and Lemma 2.3 combined together imply that r is a normal circulant. The theorem now follows from the three claims below. Claim 1 : n is an odd prime. It is obvious that n must be odd, since 2 divides n — 1. By Lemma 2.4 we have that t (n) < 3. Thus we have the following two possibilities for n: • n = p, where p is a prime; • n = p2, where p is a prime. Suppose that the latter case hold. Let A = Aut(r). Since r is a normal Cayley graph, we have A = Zn x Aut(Zn, S). We may, without loss of generality, assume that the point at infinity corresponds to the vertex 0 G Zn, and so G < Aut(Zn, S) < Aut(Zn) = Zn. Therefore, Zn contains a subgroup G of order (n — 1)/2. Since \Z*n| < n — 1 and |G| divides |Zn| we obtain that |Zn| = n - 1 or (n - 1)/2. Since, by assumption, n is not a prime, we have Z| = (n - 1)/2. This gives in the following equation p2 - 1 2 = P(P - 1) which has the unique solution p =1, a contradiction. Claim 2: n = 1 (mod 4). Since S = -S, and no element in Zn can be its own inverse, we have that the number of elements in S is even, and since |S| = n-1, we have n = 1 (mod 4). Claim 3: r is isomorphic to the Paley graph P(n). By Claim 1, n is a prime. Therefore the group Z*n is cyclic, and thus since G is a subgroup of zn, G is cyclic as well. By [6, Remark 2], we have Aut(r) = {g ^ ga + h | a e K, h e Zn}, for a suitable group K < Aut(Zn), and S is the orbit under K of a generating element of Zn, that is, S = OrbK (g) for some generating element g of Zn. Now we have that Aut(r)o = {g ^ ga + h | a e K, h e Zn : 0ff + h = 0} = {g ^ ga | a e K} = K. So we see that G < K. Since S = OrbK(g) < OrbG(g), and |S| = |OrbG(g)| we have that S = OrbG(g), which gives us that S = G (taking g = 1). Now, since G is the index 2 subgroup of the cyclic group Zn, G is of the form G = (x2} where x generates Zn. Therefore G consists of all squares in Zn and S = G, implying that r is isomorphic to the Paley graph P (n) as claimed. It is obvious that G must be cyclic, so the graph r is in fact a quasi bicirculant. □ Proof of Theorem 1.1: It follows from Theorem 3.1 and the paragraph preceding it. □ In general, if r is a vertex transitive quasi 2-Cayley graph on a group G, not isomorphic to the complete graph, then it is a strongly regular graph of a rank 3 group. Namely, the orbits of G are contained in the orbits of the stabilizer of the Aut(r)TO and since there are just two nontrivial orbits of G, then there are exactly two nontrivial orbits of the Aut(r)TO which in fact must coincide with the orbits of G. Therefore Aut(r) must be a rank 3-group, and the graphs of the rank 3 groups are strongly regular graphs. 4 Quasi 3-Cayley and 4-Cayley graphs In this section we will deal with the question which connected circulants are also quasi 3-Cayley graph or strongly quasi 4-Cayley graphs. We first consider the case of strongly quasi 3-Cayley graphs. Theorem 4.1. Let r be a connected circulant. Then r is also a strongly quasi 3-Cayley if and only if r = Cay(Zn, S) where S is the set of all non-zero cubes in Zn, and n is a prime such that n = 1 (mod 3). Moreover, r is a quasi tricirculant. Proof. Let r = Cay (Zp, S) where p = 1 (mod 3) is a prime and S is the set of all nonzero cubes in Zp. Since p is a prime, it is well known that Aut(Zp ) = Zp is a cyclic group of order p - 1. Let G = (a3}, where a is a generating element of Zp. Then G consists of all non-zero cubes in Zp, and |G| = p-1. The action of G on Zp defined by xg = g • x gives G as the subgroup of Aut(r). The group G acts quasi-semiregularly on Zp with 0 e Zp as the point at infinity. Namely, it is easy to check that G0 = G, and that the stabilizer of any element x G Zp\{0} is trivial. Since |G| = P-1, it follows that G has 3 orbits on Zp\{0}, and therefore r is a quasi 3-Cayley graph. Since one of the orbits of G is the set S, the point at infinity is adjacent to only one orbit of G, so r is in fact a strongly quasi 3-Cayley graph. By the construction r is an arc-transitive circulant since G < Aut(T)0 acts transitively on the set of vertices adjacent to the vertex 0. Conversely, let r be a connected circulant of order n, which is also a strongly quasi 3-Cayley graph on a group G. Then |G| = n-1. From Lemma 2.1 we have that r is arc-transitive, and therefore Proposition 2.1, Lemma 2.2 and Lemma 2.3 combined together imply that r is a normal circulant. Therefore, we can assume that r = Cay(Zn, S), and that G < Aut(Zn, S) < Aut(Zn), implying that n-11y(n), where ip(n) is the Euler totient function. Claim 1 : n is a prime number. Let n = PÌ1 • p2k2 ••• p\l, be a canonic factorization of a positive integer n. From Lemma 2.4, we have t(n) < 4. Now we can calculate t (n) = (ki + 1)(k2 + 1) ••• (kt + 1). We have the following possibilities for n: • n = p, • n = p2 ; 3 • n = p ; • n = pq, where p and q are different primes. If n = p2, then the only solution of 11y(n) is p =2 and n = 4. However, if n = 4, the graph r is of valency 1, so it is not a connected graph. If n = p3, then there is no solution of the above equation. If n is a product of two different primes, then we have \Z*n\ = (n - 1)/3 or 2(n - 1)/3. In the first case Z*n = G, so Z*n acts semiregularly on Zn\{0}, and it is not difficult to see that this is not the case for n = pq. If \Z*n | = 2(n - 1)/3, then we obtain the following equation ( i)( i) 2(pq -1) (p - 1)(q - 1) =-3—. The only solutions in natural numbers of the above equation are (p, q) G {(4, 7), (5, 5), (7,4)}, so there are no two different primes p, q satisfying the given equation. Having in mind all the written above, we conclude that n is a prime. Claim 2: r is isomorphic to the Cayley Graph Cay(Zn, S), where S is set of all non zero cubes in Zn, and n is a prime such that n = 1 (mod 3). Similarly as in the previous section, it can be shown that G = S. Since G is an index 3 subgroup of Zn, we have G = (x3), where x is a generating element of Z£. It follows that G consists of all cubes in Zn, so r is isomorphic to Cay(Zn, S), where S is the set of all non zero cubes in Zn and n = 1 (mod 3) is a prime. It is obvious from the mentioned above, that the group G must be cyclic, therefore, r is in fact a quasi tricirculant. □ Proof of the Theorem 1.2: Let r be a connected circulant of order n, which is also a quasi 3-Cayley on a group G. The point at infinity is adjacent to all three nontrivial orbits of G, if and only if r is isomorphic to Kn. If the point at infinity is adjacent to just one nontrivial orbit of G, then r is a strongly quasi 3-Cayley graph, therefore, Theorem 4.1 gives us the desired result. If the point at infinity is adjacent to two nontrivial orbits of G, then we consider the complement E = r of the graph r. The graph E is a quasi 3-Cayley graph on G, and actually it is a strongly quasi 3-Cayley graph on G. Since E is the complement of a circulant it is also a circulant. Suppose that E is not connected. Then, since it is vertex-transitive, it is the disjoint union of some isomorphic graphs. The point at infinity is adjacent to one orbit of G, so the connected components of E must have at least 1 + n—1 points. Therefore n = k • n\, where k is the number of connected components, and n is the number of points in each of the components. We have noticed that n > 1 + n-1, thus k < 2. If k = 1 then E is connected. Suppose that k = 2. Then there are two connected components of r, say r and r2, each containing n/2 points. Suppose that tog r^ Let Ai, A2 and A3 be a nontrivial orbits of G, and let the point at infinity be adjacent to Ai. Then Ai c ri. Since ri and r2 have the same size, it means that at least one of A2 and A3 have points both in r i and r2. Suppose that u, v g A2 , such that u g r i and v g r2. Since u and v are in the same orbit of G then there exist g g G which maps u to v. However, g fixes to, and consequently g fixes ri, a contradiction. Having in mind all the written above, we see that E is a connected circulant, which is also a strongly quasi 3-Cayley graph. Therefore we have the desired result. □ We will continue this section with the proof of Theorem 1.3. Proof of Theorem 1.3: Let r = C9. Then r = Cay(Z9, {±1}). Then the group G = {1, -1} C Zg acts quasi semiregularly on Z9 with 0 as the point at infinity. Let r = Cay(Zp, S), where S is the set of all fourth powers in Zp\{0}, and p is a prime such that p = 1 (mod 4). Define G = (a4), where a is some generating element of Zg, which is cyclic in this case. We have that G acts quasi-semiregularly on Zg with 0 as the point at infinity. Since |G| = i, it follows that G has 4 orbits on Zp\{0}, and therefore r is a quasi 4-Cayley graph. It is also easy to see that 0 is adjacent to only one orbit of G on Zp\{0}, therefore r is a strongly quasi 4-Cayley graph. By the construction, r is a connected arc-transitive circulant. Conversely, let r be a connected circulant of order n which is also a strongly quasi 4-Cayley graph on a group G. Then |G| = (n - 1)/4. Using Lemma 2.1 we have that r is arc-transitive, and so Proposition 2.1, Lemma 2.2 and Lemma 2.3 combined together imply that r is a normal circulant. Therefore, we can assume that r = Cay(Zn, S), and that G < Aut(Zn, S) < Aut(Zn), implying that n1 — |y(n). (4.1) Using Lemma 2.4 we obtain t(n) < 5. So we have the following possibilities: • n = p, • n = p2, 3 • n = p 3, 4 • n = p4, • n = pq, where p and q are different primes. If n = p2, then the only solution of (4.1) is n = 9. In this case, the valency of r is (9 - 1)/4 = 2, so r = C9. In the cases when n = p3, and n = p4 there is no prime satisfying (4.1). When n = pq, we have that (p - 1)(q - 1) = a ■ (pq - 1)/4, where a e {1, 2, 3}. If a = 1, then we have Z*n = G, so Z*n must act semiregularly on Zn\{0}, which is not the case. If a = 2, then there are no two different primes satisfying (p - 1)(q - 1) = (pq - 1)/2, and finally, when a = 3, we have that n = 5 ■ 13 is the only possibility. In this case, r is a connected arc-transitive circulant on 65 vertices, which has valency 16. Since G is an index 3 subgroup of Z65 = Z4 x Zi2, then we can calculate G = {±1, ±8, ±12, ±14, ±18, ±21, ±27, ±31}, and we can see that G does not act semiregularly on Z65\{0}. Namely, the non identity element 21 e G fixes the point 13 e Z65\{0}. Assume now that n is a prime. Similarly as in the proof of Theorem 3.1, we obtain G = S, and therefore, since G is an index 4 subgroup of Z*n, we have G = (x4), where x is some generating element of Z*n. Therefore, r = Cay (Zn, S), where S is the set of all fourth powers in Zn\{0}. From the mentioned above, it is clear that G is a cyclic group, so r is in fact a quasi tetracirculant. □ References [1] E. Dobson, A. Malnic, D. Marušic and L. A. Nowitz, Minimal normal subgroups of transitive permutation groups of square-free degree, Discrete Math. 307 (2007), 373-385. [2] E. Dobson, A. Malnic, D. Marušic and L. A. Nowitz, Semiregular automorphisms of vertex-transitive graphs of certain valencies. J. Combin. Theory Ser. B 97 (2007), 371-380. [3] M. Giudici, Quasiprimitive groups with no fixed point free elements of prime order, J. London Math. Soc. 67 (2003), 73-84. [4] M. Giudici, New constructions of groups without semiregular subgroups, Comm. Algebra 35 (2007), 2719-2730. [5] M. Giudici and J. Xu, All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism, J. Algebr. Combin. 25 (2007), 217-232. [6] I. Koväcs, Classifying arc-transitive circulants, J. Algebr. Combin. 20 (2004), 353-358. [7] K. Kutnar, A. Malnic, L. Martinez and D. Marušic, Quasi m-Cayley strongly regular graphs, manuscript. [8] K. Kutnar and D. Marušic, Recent trends and future directions in vertex-transitive graphs, Ars Math Contemp. 1 (2008), 112-125. [9] K. Kutnar and P. Šparl, Distance-transitive graphs admit semiregular automorphisms, European J. Combin. 31 (2010), 25-28. [10] C. H. Li, Permutation groups with a cyclic regular subgroup and arc-transitive circulants, J. Algebr. Combin. 21 (2005), 131-136. [11] D. Marušic, On vertex symmetric digraphs, Discrete Math. 36 (1981), 69-81. [12] M. Y. Xu, Automorphism groups and isomorphisms of Cayley digraphs, Discrete Math. 182 (1998), 309-320. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 155-170 2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs Shaofei Du School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Gareth Jones * School of Mathematics, University of Southampton, Southampton S0171BJ, United Kingdom Jin Ho Kwak Department of Mathematics, POSTECH Pohang 790-784, Korea Beijing Jiaotong University, Beijing 100044, P. R. China Roman Nedela Institute ofMathematics, Slovak Academy ofScience Severnà 5, 975 49 Banskà Bystrica, Slovakia Martin Skoviera Department of Computer Science, Comenius University, 842 48 Bratislava, Slovakia Received 31 December 2011, accepted 11 May 2012, published online 2 July 2012 Abstract We classify those 2-groups G which factorise as a product of two disjoint cyclic subgroups A and B, transposed by an automorphism of order 2. The case where G is meta-cyclic having been dealt with elsewhere, we show that for each e > 3 there are exactly three such non-metacyclic groups G with |A| = |B| = 2e, and for e = 2 there is one. These groups appear in a classification by Berkovich and Janko of 2-groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups. Keywords: Regular map, complete bipartite graph, product of cyclic groups. Math. Subj. Class.: 20D15, 05C10, 20F05 * Corresponding author. E-mail addresses: dushf@mail.cnu.edu.cn (Shaofei Du), g.a.jones@maths.soton.ac.uk (Gareth Jones), jinkwak@postech.ac.kr (Jin Ho Kwak), nedela@savbb.sk (Roman Nedela), skoviera@dcs.fmph.uniba.sk (Martin Skoviera) 1 Introduction Groups that factorise as products of isomorphic cyclic groups have been studied for over fifty years [4,7,9,10,11]. In several recent papers [5,6,13,14,15,16,17,18] these groups have emerged as an important tool for the classification of regular embeddings of complete bipartite graphs in orientable surfaces. They also arise naturally in the theory of finite p-groups, for example in the recent classification by Berkovich and Janko [1, Chapter 87] of 2-groups with a unique non-metacyclic maximal subgroup. Our aim in this paper is to demonstrate some connections between these two problems by showing that a certain class of non-metacyclic 2-groups play an important role in both situations. As a consequence, we are able to give more information and simpler presentations for some of the groups described by Berkovich and Janko. As shown in [15], the problem of classifying orientably regular embeddings of complete bipartite graphs Kn,n is closely related to that of determining those groups G that factorise as a product AB of two cyclic groups A = (a) and B = (6) of order n such that A n B = 1 and there is an automorphism of G transposing the generators a and 6. Such groups are called isobicyclic, or n-isobicyclic if we wish to specify the value of n (see [15]). We will call (G, a, 6) an isobicyclic triple, and (a, 6) an isobicyclic pair for G. A result of Ito [9] shows that an isobicyclic group G, as a product of two abelian groups, must be metabelian. In particular it is solvable, so it satisfies Hall's Theorems, which extend Sylow's Theorems from single primes to sets of primes. This fact, together with results of Wielandt [19] on products of nilpotent groups, allows one to reduce the classification of n-isobicyclic groups to the case where n is a prime power (see [13] for full details). When n is an odd prime power, a result of Huppert [7] implies that G must be meta-cyclic. When n is a power of 2, however, Huppert's result does not apply, and indeed for each n = 2e > 4 there are non-metacyclic n-isobicyclic groups. In this paper we will study all n-isobicyclic groups where n = 2e, our main goal being to give a complete description of the corresponding isobicyclic triples (G, a, 6). In order to state our main result, let us define Gi(e,f) = (h,g | h2e = g2e = 1, hg = h1+2/) (1.1) where f = 2,..., e, and G2(e; k,l) = (a, 6 I an = 6n = [62, a2] = 1, [6, a] = a26-2(an/26n/2)k, (62)a = 6-2(an/26n/2)1, (a2)b = a-2(an/26n/2)l) (L2) where n = 2e > 4 and k, l G {0,1}, with k = l = 0 when n = 4. In fact, it is easily seen that this last group G2(2; 0,0) has a simplified presentation G2(2; 0,0) = (a, 6 J a4 = 64 = [a2, 6] = [62,a] = 1, [6, a] = a262). (1.3) Our main result shows that if n = 2e then every n-isobicyclic group has one of the above two presentations. Theorem 1.1. Let G be an n-isobicyclic group where n = 2e > 4. Then either (i) G is metacyclic, and G = G1(e, f ) for some f G {2,..., e}, or (ii) G is not metacyclic, in which case either G = G2(2;0,0), or e > 3 and G = G2(e; k, l) where k, l G {0,1}. In the latter case there are, up to isomorphism, just three groups for each e, with G2(e; 0,1) = G2(e; 1,1). The metacyclic groups G1 (e, f ) were treated in detail in [5]; for instance, it was shown there that, up to automorphisms of G, one can take the isobicyclic pair to have the form a = gr and b = gr h, where r is an odd integer such that 1 < r < 2e-f. This paper is therefore devoted to the non-metacyclic groups G2 (e; k, l). These groups G2 (e; k, l) have recently arisen in a purely group-theoretic context. In [1, Chapter 87] Berkovich and Janko, having classified the minimal non-metacyclic 2-groups (i.e. those with all their maximal subgroups metacyclic), then classify those 2-groups with a unique non-metacyclic maximal subgroup. Clearly such a group requires at most three generators (two to generate a metacyclic maximal subgroup, and one more outside it). The 3-generator groups of this type are relatively easy to deal with, and Berkovich and Janko devote most of their analysis to the 2-generator groups. In Corollary 87.13 they show that any such group factorises as a product of two cyclic groups, and conversely in Theorem 87.22 they show that any non-metacyclic group which factorises in this way (and is therefore a 2-generator group) has a unique non-metacyclic maximal subgroup. Their analysis of the 2-generator groups depends on considering the different possibilities for the commutator subgroup, and one part of the classification (essentially Theorem 87.19, see also [12, Theorem 4.11]) is as follows: Theorem 1.2. (Berkovich and Janko) Let G be a 2-generator 2-group with exactly one non-metacyclic maximal subgroup. Assume that G' = C2r x C2r+i where r > 2. Then G = (a, x | a2 + =1, [a, x] = v, [v, a] = b, v2 + = b2 = [v, b] = 1, v2 r = z, b2r-1 = u, x2 e (u, z) = C2 x C2, bx = b-1, (1.4) vx = v-1, ba = b-1, a4 = v-2b-1w, w e (u, z) ) with |G| = 22r+4 and G' = (b) x (v) = C2 r x C2 r+i. One should regard (1.4) as giving sixteen presentations for each r, since there are four possibilities for each of x2 and w in the Klein four-group (u, z). In Theorem 4.2, we will show that the groups G2(e; k, l) for e > 4 are exactly those groups G in Theorem 1.2 for which x2 = zk and w = zl for some k, l e {0,1}, with e = r + 2. As noted by Janko in [12, p. 315], the classification problem is not completely solved since some pairs of presentations define isomorphic groups. Indeed Theorem 1.1 shows that for each r > 2 there are, up to isomorphism, just three groups presented by (1.4) with x2 = zk and w = zl, those with l = 1 and k = 0, 1 being isomorphic to each other. As a consequence of Theorem 4.2, in (1.2) we give slightly more transparent presentations for these groups, showing that each is an extension of its Frattini subgroup $(G) = C2r+i x C2r+i by C2 x C2: the roles of a, b and an/2bn/2 in (1.2) are played by a, ax and the central involution z in (1.4). Moreover, all our structural results proved in Section 2 for the groups G2(e; k, l) apply to these groups G. For instance, we show that they are all metabelian, of exponent 2e and nilpotence class e. In classifying all isomorphisms between the groups G2(e; k, l), we also determine their automorphisms; in particular, we show that for each e > 3, Aut G2(e; k, l) has order 24e-3 or 24e-4 as l = 0 or 1. Section 3 begins a structural analysis of isobicyclic 2-groups in general, while Section 4 is devoted specifically to non-metacyclic isobicyclic groups G. We show that if n = 2e then either G has a cyclic derived subgroup, in which case e = 2 and G = G2(2; 0,0), or G has a derived group generated by two elements, in which case e > 3 and G is isomorphic to one of the three non-isomorphic groups of the form G2(e; k, l). This proves part (ii) of Theorem 1.1, and since part (i) is dealt with in [5], it completes the proof of that theorem. In Section 5 we apply results from the preceding sections to the classification of regular embeddings of complete bipartite graphs Kn,n where n is a power of 2. A completely different proof of Theorem 1.1(ii) has already been given in [6]; it proceeds by induction on e, based on the fact that if n = 2e then any n-isobicyclic group has an m-isobicyclic quotient where m = 2e-1. However, the main purpose of that paper was not to study these groups for their own sake, but rather to enumerate them and to obtain sufficient information about them to determine the corresponding graph embeddings. Here we present an alternative proof, designed to shed more light on the internal structure of these groups, and on how they are related to more general classes of 2-groups. 2 Non-metacyclic groups G2 ( e ; k, I ) In this section we analyse properties of the non-metacyclic groups G2 (e; k, l) appearing in Theorem 1.1. Throughout this section we write G(k, l), or simply G, instead of G2(e; k, l). For brevity we also write n = 2e and m = n/2 = 2e-1. It is useful to note that each group G has a Frattini subgroup $ = $(G) = (a2)x(b2) = Cm x Cm, with G/$ = C2 x C2 (see [6, Prop. 2.1]). It therefore has three maximal subgroups, namely ($, a) = $ U $a, ($, b) = $ U $b and ($, ab) = $ U $ab. Lemma 2.1. The following properties hold in G = G(k, l). (i) The elements am, bm, and z = ambm are central involutions of G. (ii) j at = albj, for i and j even, alb-jzj/2, for i odd and j even, a-lbjzll/2, for i even and j odd, a-lb-jzk+l(l+j)/2, for i and j odd. (iii) The element g = a®bj has order |g| dividing m, for i and j even, equal to n, with gm = am, for i odd and j even, equal to n, with gm = bm, for i even and j odd, dividing 4, and equal to 2 if k = l = 0, for i and j odd. (iv) The group G is isobicyclic, that is, G = (a)(b), where |a| = |b| = 2e and (a) n (b) = 1, and there is an involutory automorphism of G interchanging a and b. (v) G' = (a2b-2zk) x (aV) with (a2b-2zk) = Cm and (aV) = Cm/2. (vi) G is not metacyclic. (vii) G has nilpotence class e, with upper central series 1 = Z0 < Z1 < • • • < Ze = G where Z, = (a2e-i)(b2e-i) for i = 0,1,..., e. Proof. Some of these results were proved in [6]; for completeness we give proofs here. If we define z = ambm, the defining relations for G in (1.2) take the form an = bn = [b2, a2] = 1, [b, a] = a2b-2zk, (b2)a = b-V, (a2)b = a-V, (2.1) where k, l e {0,1} with k = l = 0 if n = 4. (i) Since m is even, and [a2, b2] = 1, the involutions am and bm commute; they are distinct, so their product z is also an involution. Since z commutes with a2 and b2, and either m is divisible by 4 or l = 0, we have (bm)a = ((b2)a)m/2 = (b-2zl)m/2 = b-mzlm/2 = b-m = bm, so bm is an element of the centre Z (G) of G. Similarly, am e Z (G), so z e Z (G). (ii) Now we compute Va1. Define c = [b, a] = b-1a-1ba. If both i and j are even, then bj a1 = aibj. If i is odd and j is even, then since i — 1 is even we have bj ai = (bj ai-1)a = ai-1bj a = ai((b2)a)j/2 = aib-j zlj/2. If i is even and j is odd, then bj ai = b-1(bj+1ai) = (b-1aib)bj = ((a2)b)i/2bj = a-ibj zli/2. If both i and j are odd, then bj ai = bj-1abcai-1 = bj-1ab(b-2a2 zk)ai-1 = a(bj-1)a(ai+1)bb-1zk = ab1-j zl(j-1)/2a-i-1zl(i+1)/2b-1zk = a-ib-j zk+l(i+j)/2. (iii) If i and j are both even then (ii) implies that g2 = (aibj)2 = ai(bjai)bj = a2ib2j e {a4) x (b4) = Cm/2 x Cm/2, so gm = 1. If i is odd and j is even then (ii) gives g2 = ai(bj ai)bj = a2ib-j zlj/2bj = a2izlj/2, so (g4) = (a4i) = (a4); thus (g2) = (a2") for all r > 2, so |g| = |a| = n with gm = am. The proofs in the other two cases are similar. (iv) The formula in (ii) show that every element of G can be expressed in the form aibj, so G = (a)(b). In order to see that (a) n (b) = 1, note that a and bj lie in distinct cosets of $ unless i and j are both even; in this case the fact that $ = (a2) x (b2) ensures that (a) n (b) = 1. The defining relations of G are equivalent to those obtained by transposing a and b, so this transposition can be extended to an automorphism a of order 2 of G. Hence G is an n-isobicyclic group. (v) Since G = (a, b), G' is the normal closure (c9 | g e G) in G of the commutator c = [b, a]. We will show that this is the subgroup M := (c, ca). Since c = [b, a] = a2b-2zk we have ca = (a2b-2zk)a = a2b2zk+l, and conjugation by a transposes these two generators of M since [c, a2] = 1. Similarly, conjugation by b transposes the generators c and (ca)-1 of M, so M is normal in G and hence M = (c9 | g e G) = G'. Thus G' has generators c = a2b-2zk = akm+2bkm-2 and cac = a4zl = alm+4blm; these generate disjoint cyclic groups of orders m and m/2, so G' = (a2b-2zk) x (a4zl) = Cm x Cm/2. (vi) For e > 3 the fact that G' is not cyclic immediately implies that G is not metacyclic. In the case e = 2 it is easily seen that the only cyclic normal subgroups of G are contained in $, and these do not have cyclic quotients. (vii) This follows by induction on e, using the facts that Z (G) = {1 ,am,bm,z} (a simple consequence of (ii)), that G/Z(G) = G(e-1; 0,0), and that G(2; 0,0), as presented in (1.3), clearly has class 2. □ Proposition 2.2. Each isomorphism a : G(kl,ll) ^ G(k, l) is given by setting al = aibj and bl = af bh, where (i) ki = k + l(f+hp-j) (mod 2), (ii) li = l, and (iii) either i and h are odd and j and f are even, or i and h are even and j and f are odd. Moreover, each choice of the parameters i, j, f and h satisfying the above conditions determines an isomorphism G(ki, li) ^ G(k, l). Proof. Recall that G = G(k, l) = (a, b I an = bn = [b2, a2] = 1, [b, a] = a2b-2zk, (b2)° = b-V, (a2)b = a-V) with z = ambm, and define Gi = G(ki,li) = (ai,bi I an = b? = [bi, ai] = 1, [b1,a1] = a2b-2z1fcl, (b1)ai = b-2i11, (a2)bi = a-2z11 ), where z= = af bf. An isomorphism a : G= ^ G is uniquely determined by an assignment ai ^ a2 = a®bj, bi ^ b2 = af bh for some integers i, j, f and h such that a2 and b2 generate G and satisfy the defining relations of G1, when substituted for a1 and b1. Now ai has order n, whereas Lemma 2.1(iii) shows that a2 has order less than n if i and j are both even or both odd. We may therefore restrict attention to mappings a for which i and j have opposite parity, that is, a2 G $a U $b. A similar argument shows that b2 G $a U $b. If a2 and b2 are both in $a, or both in $b, they are both contained in a maximal subgroup $ U $a or $ U $b of G and hence cannot generate G. They therefore lie in distinct cosets $a and $b, and by composing a with the automorphism a of G transposing a and b if necessary, we may assume that a2 G $a and b2 G $b, that is, i and h are odd while j and f are even. This ensures that a2 and b2 generate G, since none of the three maximal subgroups of G contains both of them. For any g G G we have g2 G $ = Cf x Cf, so a2 and b2 satisfy the first three relations a? = bn = [b2,a2] = 1 for Gi. Now a sends z= = af bf to af bf. Since i is odd and j is even, we have af = am by Lemma 2.1(iii). Similarly bf = bf, so a sends z= to af bf = z. We can now consider the fourth relation. Straightforward calculations give [b2,a2] = [af bh,aV ] = a2ib-2hzk+1(h-i)/2 and so we require i26-2 zfcl = a2ib-2hzkl+l(j-f )/2, kl = k + l(f + h - i - j) (mod 2), giving condition (i) of the Lemma. For the fifth relation, we have (b2r = ((af bh)2)°V = ((b2hz1f/2)oV = b-2hz'+1f/2 and b2-Vi = b-2hzli-lf/2; since f is even and h is odd we require h = l. Similar arguments show that the sixth and final relation is also equivalent to this, so we have condition (ii). Conditions (i) and (ii) are necessary and sufficient conditions for a to be an isomorphism, in the case where a2 G $a and b2 G $b, that is, i and h are even while j and f are odd. For the case where a2 G $b and b2 G $a we can compose a with a, transposing i with j, and f with h; this gives condition (iii) of the Lemma, leaving conditions (i) and (ii) unchanged. □ Corollary 2.3. For each e > 3 we have G(1,1) = G(0,1) while G(0,0), G(1,0) and G(0,1) are pairwise non-isomorphic. Proof. From Proposition 2.2 we immediately deduce that G(k, 0) ^ G(k', 1) for any k and k', and that G(0,0) ^ G(1,0). Furthermore, taking i = 3, j = f = 0 and h =1 in the definition of a, we get an isomorphism from G(0,1) to G(1,1). □ Corollary 2.4. The automorphisms of G(k, l) are given by a : a ^ aibj, b ^ af bh where (i) either i and h are odd and j and f are even, or i and h are even and j and f are odd, and (ii) i + j = f + h (mod 4) if l = 1. Proof. This follows immediately from Proposition 2.2, with ki = k and l\ = l. □ By counting choices of i, j, f, h G Zn satisfying the conditions of Corollary 2.4, we deduce that |Aut G(k, l)| = n4/8 or n4/16 as l = 0 or 1. 3 The derived group of an isobicyclic 2-group In this section we begin an analysis of the structure of an isobicyclic 2-group. Let (G, a, b) be an n-isobicyclic triple where n = 2e > 4. As before, let c = [b, a] and let $ denote the Frattini subgroup $(G) of G. Let öi(G) = (g2' | g G G), and let Ki(G) = [G, G, • • • , G] (i times); in particular, K2(G) = G'. Each of these subgroups $(G), Oi(G) and Ki(G) is a characteristic subgroup of G. The following properties of G follow from more general known results. Lemma 3.1. Let (G, a, b) be a non-abelian n-isobicyclic triple where n = 2e > 4, and let A = (a) and B = (b). Then the following hold. (i) The derived group G' is abelian (see [9]). (ii) G'/(G' n A) is isomorphic to a subgroup of B (see [3, Corollary C]). (iii) G is metacyclic if and only if G/$(G')K3(G) is metacyclic (see [2] or [8, Hilfssatz III.11.3]). Lemma 3.2. [5, Lemma 3.1] Let G be an isobicyclic 2-group of exponent 2e > 4. Then G has a central series 1 = Z0 < Z1 < Z2 < ■ ■ ■ < Ze = G of subgroups Zi = {a2*-)(b2e-) of order 22i. Moreover, Oi(G) = Ze_i and Zi/Zi_i = C2 x C2 for each i G {1,2,..., e}. In particular, for every element g G Zi we have |g| < 2i. Outline proof. We proved this result as Lemma 3.1 of [5], so we simply outline the argument here. By a result of Douglas [4] and Ito [10] (see also [8, VI.10.1(a)]), the core of A in G is nontrivial. Since (a2* ) is the unique minimal normal subgroup of A it is therefore normal in G, and hence central. The same applies to (b2 ), so these two disjoint subgroups generate a central subgroup Z1 = C2 x C2. Now apply the same argument to the isobicyclic group G/Z1, and iterate. □ Lemma 3.3. Let (G, a, b) be a non-abelian n-isobicyclic triple where n = 2e > 4, and let A = (a) and B = (b). Then G has the following properties. (i) There exist an odd integer d < 2e and integers u and v such that 0 < u < v < e, c = [b,a] = ad2" b_d2" and G' = (c) x (a2" ) = (c) x (b2" ). In particular, G' is cyclic if v = e. Moreover, [a2", b2" ] = [b2", a2" ] = 1. (ii) (c) n A = (c) n B = 1, |c| = 2e_u, and for each integer j such that 0 < j < e — u there exists an odd integer h such that c2j = ah2"+j b_h2"+j. (iii) Either G' = (c), or G' = (c, ca) = (c, cb) with ca = csat2" and cb = csb_t2" where s and t are odd. Proof. Since G = AB, each element can be written as aibj, and since A n B = 1 this representation is unique. Let c = [b, a] = ar bw. Since the automorphism a interchanges a and b, we have braw = [ba, aa] = [a, b] = [b,a]-1 = b_wa_r. Therefore w = —r (mod 2e) and so c = arb_r. We can write r = d2u where d is odd, d < 2e and 0 < u < e. Similarly, for every integer j there is an integer k such that cj = akb_k. In particular, (c) n A = (c) n B = 1, as claimed in (ii). Let the cyclic group G' n A be generated by a2", where v < e. Applying» a gives G' n B = (b2"). Since G = (a, b), Lemma III.1.11 of [8] implies that G' = (cg | g G G). By Lemma 3.1(i), G' is abelian, so c is an element of G' of maximal order. Since (c) n A = (c) n B = 1, we see that (c) x (a2" ) < G'. By Lemma 3.1(ii), G'/(a2" ) is cyclic. Since (c) n A =1 again, the image of c in G'/(a2" ) has order |c|, so it is an element of G'/(a2" ) of maximal order and therefore generates G'/(a2" ). This gives G' = (c) x (a2" ) and hence, by applying a, G' = (c) x (b2" ). From a2" c = ca2" and c = ad2" b_d2" we see that [bd2", a2" ] = 1. Hence it follows that [(bd2"), a2"] = 1, and in particular, since d is odd, [b2", a2"] = 1. By symmetry, [a2" ,b2" ] = 1. Since (c) = (ad2"b_d2") < Ü„(G) = Ze_„, Lemma 3.2 shows that |c| < 2e_u. Since c is an element of maximal order in G', we have 2e_v = |a2" | < |c| < 2e_u, so u < v. This proves (i), apart from the inequality u = v, which follows later. To prove (ii), let L denote the subgroup G'(b2"}. Then L = (ad2" b-d2" , b2" }(b2"} = (a22 , b2"} = (a2" }(b2"} = Ze-u. Computing the order 22(e-u) = |L| = |G'||(b2"}|/|G' n (b2"}| = |c|2e-v2e-u/2e-v = |c|2e-u, we see that |c| = 2e-u. For each j = 0,1,..., e — u we have c2j G Ze-(u+j-), so c2j = ah2"+j b-h2"+j for some integer h. Since |c2j | = 2e-(u+j), it follows that c2j G Ze-(u+j-+1), so h is odd. This proves (ii). We now consider (iii). Since ca G G' = (c} x (a2"}, either ca G (c}, or ca = csat2q for some integers s, t and q where t is odd and q > v. In the former case we have G' = (c}, satisfying (iii); we may therefore assume the latter, in which case we also have cb = csb-t2q. Define M = (c, ca}, so M = (c} x (a2"}. From the preceding paragraph we know that c2" " = ah2" b-h2" for some odd h. Therefore b2" g (bh2q} = (c-2"-" ah2q} < (c2"-", a2"} < M, which implies that M = (c} x (b2"}. Now Ma = (c, a2" }a = M and Mb = (c, b2" }b = M, so Mg = M for each g g G. In particular, cg G M for each g G G. Therefore G' = (cg j g G G} = M. In other words, q = v, that is ca = csat2" where t is odd. We now show that u = v. Recall that G' = (c} x (a2"} = (c} x (b2"}, so Ze-v = (a2"} x (b2"} < G'. On the other hand c = ad2"b-d2", so G' < (a2" }(b2"} = Ze-u. Now suppose that u = v, so G' = Ze-v, with e — v > 0 since G is non-abelian. By Lemma 3.2, the subgroup G'/Ze-(v+1) = Ze-v/Ze-(v+1) is central in G/Ze-(v+1). We have seen that G' = (c, ca}, so caZe-(v+1) = cZe-(v+1) since c G G'. Thus G'/Ze-(v + 1) = ca}/Ze-(v+1) = (c}/Ze-(v+1) is cyclic, contradicting the fact that Ze-v/Ze-(v+1) = C2 x C2 by Lemma 3.2. Thus u < v, completing the proof of (i). Finally, since ca = csat2" G G' = (c} x (a2"} we have 2e-u = |c| = |ca| = |csat2" | = max{|cs|, |at2" |}, with |at2" | = 2e-v < 2e-u since v > u, so |cs | = |c| and hence s must be odd. □ The next result uses the parameter u, where c has order 2e-u, to distinguish between metacyclic and non-metacyclic n-isobicyclic groups G. Lemma 3.4. Let (G, a, b) be a non-abelian n-isobicyclic triple where n = 2e > 4. With u and v defined as in Lemma 3.3, the following statements hold. (i) If u > 2 then G is metacyclic, with 2 < u < v = e and G' = (c} = ((ab-1)r} = (ii) If u < 2 then G is non-metacyclic, with u = 1, v = 2 and G' = (a2b-2} x (a4} = (a2b-2} x (b4}, where (a2b-2} = C2.-i and (a4} = (b4} = C2e-2. In particular, if G is non-metacyclic and G' is cyclic, then e = 2 and G' = (a26 2) = C2. Proof. By Lemma 3.3, G' = (c) x (a2" ) where c = [6, a] = ar6-r with r = d2" for an odd integer d and some integers u and v such that 0 < u < v < e. (a) We first consider the case where G' is cyclic, so that v — e. By Lemma 3.3, G' — (c) and hence ca = cs for some s, which must be odd. By applying a, which inverts c, we also have cb = cs. It follows that cab = c. Moreover, [c, a] = [c, 6] = cs-1 G (c2), which means that the image c(c2) of c in G/ (c2) is a central involution in that group. (As a characteristic subgroup of (c) = G', (c2) is normal in G.) Now we have (a6-1)2 = a6-1a6-1 = a2(a-16a)-16-1 = a2(6c)-16-1 = a2c-16-2 = a26-2c (mod (c2)) and a262 = a6ac-16 = 6ac-1 ac-16 = 6aa6 = 6a6ac-1 = 62ac-1ac-1 = 62a2 (mod (c2)). (a1) Suppose that u > 2, as in (i). Then r/2 is even, so (a6-1)r = ((a6-1)2)r/2 = (a26-2c)r/2 = (a26-2)r/2cr/2 = (a26-2)r/2 = ar6-r = c (mod (c2)). Thus (a6-1)r is an odd power of c, so ((a6-1)r) = (c) and |(a6-1)r | = |c| = 2e-" by Lemma 3.3. Since G = (a, 6), the quotient group G/G' = G/(c) is generated by the images a and a6-1 of a and a6-1 in this group. Now (c) n A =1 by Lemma 3.3(ii), so a has order |a| = 2e. Since (a6-1)r G (c), we see that a6-1 has order dividing r, and hence dividing 2". But G/(c) is an abelian group of order |G|/|(c)| = 22e/2e-" = 2e+", so a6-1 must have order 2" with G/G' = (a) x (a6-1 ) = C2e x C2u. Since (a6-1)r is an odd power of c we have ((a6-1)r) = (c), so the cyclic subgroup H := (a6-1) contains G' with index 2". Since the image of H in G/G' has order 2", and G' has order 2e-", it follows that H has order 2e. Since H contains G' it is a normal subgroup of G. Thus AH = HA is a subgroup of G, and since it contains both a and 6 we have G = AH, so G is metacyclic. This proves (i) in the case where G' is cyclic. (a2) Now suppose that G' is cyclic and u = 0. Then G' = (c) has order 2e-" = 2e by Lemma 3.3(ii). Since G'nA = 1 by Lemma 3.3(ii) we have |G'A| = |G'||A| = 22e = |G|, so G = G'A and G/G' is cyclic. But then G/$ is cyclic and hence so is G, a contradiction. Hence u = 0. We therefore have u = 1, so r = 2d, giving c = a2d6-2d, where d is odd. By Lemma 3.3(ii), |c| = 2e-1, and since G = (a, a6-1) we have G/G' = G/(c) = C2e x C2. (a3) Suppose first that G is metacyclic. Huppert gives the general form for a metacyclic p-group in [8, III.11.2]; taking p = 2 we have G = (g, h | h2' = 1, g2' = h2', hg = hq) with 0 < k < i, q2' = 1 (mod 2®) and 2k(q — 1) = 0 (mod 2®). Thus G has anormal subgroup H = (h) = C2i with G/H = C2j, so |G| = 2®+j and hence i + j = 2e. Since |h| =2® and G has exponent n = 2e we have i < e and hence j > e. Since |g| = 2®+j-k we have i + j — k < e and hence k > e. But k < i, so i = j = k = e. Thus G = (g, h | gn = hn = 1, hg = hq) for some q satisfying 1 < q < n. Now G', being cyclic, is generated by [h, g] = hq-1. We are assuming that u = 1, so G' = C2e-i and hence q = 3 (mod 4). Each element of G has the form g®hj for a unique pair i, j e Zn. By using the relation (hj)gi = hjqi, we obtain (gžhj)m = gim(hj)si(m-i) (hj)si(m-2) ... (hjKhj = gimhj(qi(m-i)+qi(m-2) + - + qi+1) = gimhj(qim-1)/(9i-1) for all m > 1. Let m = n/2 = 2e-1. If i is even then gim = 1 and ql = 1 (mod 4); if 2k II qi - 1 then 2k+e-1 || qim - 1, so 2e-1 || (qim - 1)/(qž - 1) and hence (gžhj)m = hjm. If i is odd then gim = gm and qi = 3 (mod 4); if 2k || qi + 1 (so k > 2) then 2k+e-1 || qim - 1, and 2 || qi - 1, so 2e | (qim - 1)/(qi - 1) and (gihj)m = gm. Thus (gi hj )m hjm, for i even, gm, for i odd, so gihj has order n if and only if i or j is odd, that is, gihj e $ = (g2, h2). If a and b are an isobicyclic pair for G then they have order n, so they are not elements of Since they generate G, they are in different cosets of namely g$, h$ or gh$. The subgroups A = (a) and B = (b) are disjoint, so am = bm; hence these two cosets cannot be g$ and gh$ (otherwise am = gm = bm), so one of them must be h$, say a e h$. Then = Hso H= B$ = g$ or gh$, giving (ha)m = gm = hm and hence Ha n H =1. Since H is a normal subgroup of G, so is Ha. Hence G = Ha x H, which is abelian, contradicting the assumption. Thus G cannot be metacyclic. (a4) Now suppose that G is non-metacyclic, with G' cyclic and u = 1 as before. We consider the subgroup N := (c, ab-1) of G; this is abelian since cab = c, and it is normal in G since it contains G' = (c). Note that N is the preimage in G of (ab-1) < G/G'. In the abelian group G/G' = G/(c) we have (ab-1)2d = a2db-2d = c =1, which means that ab-1 is of order 2. Since |c| = 2e-1, we have |N| = 2e. Since (N, a) = (a, b) = G, we deduce that G = N x (a). Since G is not metacyclic, N can not be cyclic and so N = C2e-1 x C2. Let c' be an involution of N different from c2e , so that N = (c) x (c'). Then the conjugacy action of a on N is defined by ca = cs and (c' )a = cj2e- c', where j = 0 or 1, and s is odd. Now G = (c, c', a) with [c, c'] = 1, so G' = ([c, a]g, [c', a]g | g e G) < (c2, cj2e 2). Since G' = (c), we see that j2e-2 must be odd, so j = 1 and e = 2, giving |G| = 16. Since v = e we have v = 2, and we have proved (ii) in the case where G' is cyclic. (Note that a4 = b4 = 1 in this case.) (b) We now consider the case where G' is not cyclic, that is, v < e. This immediately implies that G is not metacyclic. Recall that u < v. Since G' = (c) x (a2"), we have $(G') = (c2)x(a2"+i). Moreover, by Lemma 3.3(iii) we have ca = csat2" for some odd integers s and t. Then at2" = c-s+1[c, a] e L := $(G')K3(G), which implies that a2" e L. Since L is a characteristic subgroup of G, it also contains b2" = (a2" )a, and hence it contains the subgroup Ze-v = (a2e " )(b2e " ). Suppose that G/Ze-v is metacyclic. Since G/L = (G/Ze-v)/(L/Ze-v), it follows that G/L is metacyclic. Then Lemma 3.1(iii) implies that G is metacyclic, which is a contradiction. Therefore G/Ze-v is non-metacyclic. Now G/Ze-v is an isobicyclic 2-group. Since it is non-metacyclic, and its derived group (G/Ze-v)' = G'/Ze-v = C2v-u is cyclic, it follows from part (a2) of this proof that G/ze-v has order 16, with |G'/Ze-v | = 2. Thus a4 G Ze-v = (a2")(b2"), so v = 2. Since G'/Ze-v = C2v-u and |G'/Ze-v | = 2, we deduce that v - u = 1, so u = 1. We have G' = (c) x (a2" ) = (c) x (b2" ) with c = ad2" b-d2" = a2db-2d for some odd d, so G' = (a2b-2) x (a4) = (a2b-2) x (b4), with first and second factors cyclic of orders 2e-1 and 2e-2 as required for (ii). The final statement in the Lemma is an immediate consequence of (i) and (ii). □ 4 Non-metacyclic isobicyclic 2-groups The following theorem characterises non-metacyclic isobicyclic 2-groups. Theorem 4.1. Let (G, a, b) be a non-metacyclic n-isobicyclic triple with n = 2e > 4. (i) If e = 2 then G = (a, b I a4 = b4 = [a2, b] = [b2, a] = 1, [b, a] = a2b2) = G2(2;0, 0). (ii) If e > 3 then G = (a, b I an = bn = [b2,a2] = 1, [b, a] = a2b-2(an/2bn/2)k, -2(an/2bn/2)l (a^b _ „-2(„n/2un/2\l\ (b2)a = b-2(an/2bn/2)', (a2)b = a-2(an/2bn/2)') = G2(e; k,l) where k, l G {0,1}. Proof. (i) If e = 2 then c = a2b-2 = a2b2 is an involution commuting with both a and b, so c is central in G. Thus [a, b2] = [b, a2] = 1 and it follows that G = G2(2; 0,0). (ii) Let e > 3. By Lemma 3.4(ii) we see that v = 2 and u = 1, so G' = (c) x (a4) = (a2b-2) x (a4) = C2e-i x C2e-2 where c = [b, a] = a2db-2d for some odd d. By Lemma 3.3(iii) we have ca = csa4t and cb = csb-4t (4.1) for some odd s and t. We will determine d, s and t up to group automorphisms. By Lemma 3.4(ii), [a4, b2] = [a2, b4] = 1; since d — 1 is even, this implies that ca2b-2 = b2a-2a2db-2da2b-2 = b2 a2(d-1)b-2da2b-2 = a2(d-1)b-2(d-1)a2b-2 = a2db-2d = c, a2 b2 so ca = cb . Since ca2 = (ca)a = (csa4t)a = (ca )sa4t = cs2 a4t(s+1) and (cb)b = (csb-4t)b = (cb)sb-4t = cs2 b-4t(s+1), 2 b c we see that a4t(s+i) = b-4t(s+i). However, A n B = 1, so a4t(i+s) = 1 and hence c°2 = cs . Since t is odd, we have s = -1 (mod 2e-2). In what follows, we set s = -1 + l2e-2; since |c| = 2e-i we can assume that l = 0 or 1. Then s2 = 1 (mod 2e-i), and because |c| = 2e-i we have ca = cs = c. Thus a2 commutes with c = a2db-2d, and hence with b2d ; since d is odd we therefore have [a2, b2] = 1. (4.2) Using equation (4.1) we see that b-ia2j b = ((ab)2)j = ((ac-i)2)j = (ac-iac-i)j = (a2(ca)-ic-i)j = (a2a-4tc-(s+i)j = a2(i-2t)j c-j(s+i) (4.3) for each positive integer j. By taking j = 2e 2 we deduce that the involution a2° 1 is central in G, and the same holds for b2° . In what follows we set z = a2° b2° = c2° . By equations (4.1) and (4.3) we have ca = c-ia4V, cb = c-ib-4V, b-ia2j b = a2(i-2t)j z1j. (4.4) From cb = (a2db-2d)b and equation (4.4) we have a-2db2db-4V = a2(i-2t)dz1db-2d, so and hence a4(i-t)d = b4(d-t) (1 - t)d = d - t = 0 (mod 2e-2). Solving these equations gives d = t = 1 (mod 2e-2). Writing d = 1 + k2e-2 where k = 0 or 1, and using b2e = 1, we see from these two congruences that the relations c = a2db-2d, ca = csa4t and cb = csb-4t can be respectively rewritten as [b, a] = a2+k2e-1 b-2+k2e-1, (b2)a = a12e-1 b-2+12e-1, (a2)b = a-2+12e-1 b12e-1 (4.5) where k, l G {0,1}. By combining the relations in (4.5) with the fact that a2e = b2e = [a2, b2] = 1, we see that G satisfies all the defining relations of G2(e; k, l) in (1.2). Thus G is an epimorphic image of G2(e; k, l), and since these two groups have the same order, they are isomorphic. □ Recall that Theorem 1.2, of Berkovich and Janko, states that a 2-generator 2-group with exactly one non-metacyclic maximal subgroup, and with a derived group isomorphic to C2r xC2r+1 for some r > 2, has a presentation of the form (1.4). Here we consider a subset of these groups, namely those for which x2 and w are powers of the central involution z. For each r > 2, and for each pair k, l G {0,1}, let G = G(k, l) denote the group given by the presentation (1.4) with x2 = zk and w = z1, that is, G(k, l) = (a, x | a2r+2 = 1, [a, x] = v, [v, a] = b, v2r+1 = b2r = [v, b] = 1, v2r = z, b2r-1 = u, x2 = zk, bx = b-i, (4.6) vx = v-i, ba = b-i, a4 = v-2b-iw, w = z1}. Our aim is to show that G is isomorphic to the group G2 = G2 (e; k, l), where e = r + 2. To avoid notational confusion, let us present G2 as G2(e; k,1) = (ai,6i | a2e = bf = [b2,a2] = 1, [bi,ai] = a2b-2(a2e- )k, (1)ai = b-V"1 b1e-1 ), (a?)bi = a-V"1 b2e-1 )). (4.7) Theorem 4.2. For each e = r + 2 > 4 there is an isomorphism from G(k, l) to G2(e; k, l) sending the generators a and x of G(k, l) to ai and a-ibi in G2(e; k, l). Proof. We first show that the map a ^ ai, x ^ a-ibi extends to a homomorphism G ^ G2. We map the other elements of G appearing in (4.6) into G2 by v ^ a-2bizk, b ^ b-4z^, z ^ zi, u ^ b2e 1 and w ^ z[ where zi = a2e 1 bfe 1. We need to show that the defining relations for G in (4.6) are satisfied when a, x, v, b, z, u and w are replaced with their images in G2. This is a routine matter, using the properties of G2 proved in Section 2, so we will simply illustrate it in a typical case, namely the relation x2 = zk. For this we need to show that (a-ibi )2 = zf in G2. Using Lemma 2.1(ii) and the fact that z i is in the centre of G2, we have (a-^ i )2 = a- i (b i )b i = a- i (a i zf)bi = zf, as required. The other cases are similar, so the mapping extends to a homomorphism 0 : G ^ G2. This is an epimorphism since a i and a- i b i generate G2. We proved in [6, Prop. 2.1] that |G21 = 22e, so |G| > 22e. The defining relations v2r+1 = b2" = [v, b] = 1 for G show that (v, b) has order at most 22r+i. The relations x2 = zk (= v2rk), vx = v- i, bx = b- i show that (v, b) is a normal subgroup of index at most 2 in (v, b, x), so the latter group has order at most 22r+2. Finally the relations a4 = v-2b- iz1, ba = b- i, va = vb, xa = xv- i show that (v, b, x) is a normal subgroup of index at most 4 in (v, b, x, a) = G, so |G| < 22r+4 = 22e. Thus |G| = |G2|, so 0 is an isomorphism. □ This confirms the assertions in [1, 12] that the groups G(k, l) have order 22r+4, a fact which is not immediately apparent from the presentation (4.6). 5 Regular embeddings of Kn,n where n = 2e A map M is a cellular embedding of a connected graph K in a closed orientable surface. It is (orientably) regular if the group Aut(M) of all orientation-preserving automorphisms of the embedding acts regularly on the oriented edges (darts) of K. It was shown in [15, Section 2] that every regular embedding M of a complete bipartite graph Kn n determines an n-isobicyclic triple (G, a, b). Here G is the subgroup Aut0(M) of index 2 in Aut(M) leaving the bipartition of Kn,n invariant. The generators a and b rotate a chosen edge e = uv around its incident vertices u and v to the next edge, following the orientation of the surface around u or v. The automorphism of G transposing a and b is induced by conjugation by the map automorphism reversing e. Conversely, every n-isobicyclic triple (G, a, b) arises in this way, with (G1, a1, b1) and (G, a, b) giving isomorphic maps if and only if there is an isomorphism G1 ^ G sending a1 to a and b1 to b (see [15] or [14, Proposition 2]). Thus an isobicyclic group G may have inequivalent pairs a, b leading to non-isomorphic maps. The following characterisation of regular embeddings of Kn n, where n = 2e and Auto(M) is non-metacyclic, was proved in [6]. Here we give a different proof, using the structure of non-metacyclic isobicyclic 2-groups described in earlier sections. Theorem 5.1. For each n = 2e > 8 there are exactly four non-isomorphic regular embeddings M of Kn,n for which Aut0(M) is non-metacyclic; these correspond to the four isobicyclic triples (G, a, b), where G = G2(e; k, l) and k, I G {0,1}. There is exactly one regular embedding M of K4 4 for which Aut0(M) is non-metacyclic; this map corresponds to the isobicyclic triple (G, a, b) where G = G2 (2; 0,0). Proof. If e = 2 the result follows directly from Theorem 4.1(i). We may therefore assume that e > 3, so by Theorem 4.1(ii) there are at most four isomorphism classes of isobicyclic triples, corresponding to the four presentations G2(e; k, l) where k, l G {0,1}. By Corollary 2.3, the groups G2(e; 0,0), G2(e; 1,0) and G2(e; 0,1) are mutually non-isomorphic, and hence so are the corresponding isobicyclic triples. To complete the classification it is enough to show that the triples corresponding to the isomorphic groups G2(e; 0,1) and G2(e; 1,1) are not equivalent. If there is an isomorphism from G2(e; 1,1) = (a1}(b1} to G2 (e; 0,1) = (a}(b} taking a1 to a and b1 to b then condition (i) of Proposition 2.2 gives 1 = k1 = k = 0 (mod 2), a contradiction. Hence there are four non-isomorphic maps, as claimed. □ Acknowledgements The authors acknowledge the support of the following grants. The first author was partially supported by grants NNSF(10971144), BNSF(1092010) and APVV SK-CN 022109. The third author was partially supported by grant NRF(K20110030452) of Korea. The fourth and the fifth authors acknowledge partial support from the grants APVV-0223-10, VEGA 1/1085/11, and from the grant APVV-ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation. The fifth author was partially supported by the grant VEGA 1/1005/12. The authors are also grateful to the organisers of workshops at Capital National University, Beijing, and the Fields Institute, Toronto, where much of this research took place, and to the referees for their helpful comments on the presentation of this paper. References [1] Y. Berkovich and Z. Janko, Groups of Prime Power Order. Vol. 2, Walter de Gruyter, Berlin, 2008. [2] N. Blackburn, On prime power groups with two generators, Proc. Cambridge Phil. Soc. 54 (1958), 327-337. [3] M. D. E. Conder and I. M. Isaacs, Derived subgroups of products of an abelian and a cyclic subgroup, J. London Math. Soc. 69 (2004), 333-348. [4] J. Douglas, On the supersolvability of bicyclic groups, Proc. Nat. Acad. Sci. US 49 (1961), 1493-1495. [5] S. F. Du, G. A. Jones, J. H. Kwak, R. Nedela, and M. Skoviera, Regular embeddings of Kn,n where n is a power of 2. I: Metacyclic case, European J. Combin. 28 (2007), 1595-1609. [6] S. F. Du, G. A. Jones, J. H. Kwak, R. Nedela, and M. Skoviera, Regular embeddings of Kn,n where n is a power of 2. II: The non-metacyclic case, European J. Combin. 31 (2010), 19461956. [7] B. Huppert, Über das Produkt von paarweise vertauschbaren zyklischen Gruppen, Math. Z. 58 (1953), 243-264. [8] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1979. [9] N. Itö, Über das Produkt von zwei abelschen Gruppen, Math. Z. 62 (1955), 400-401. [10] N. Ito, Über das Produkt von zwei zyklischen 2-Gruppen, Publ. Math. Debrecen 4 (1956), 517-520. [11] N. Itö and A. (Ohara, Sur les groupes factorisables par deux 2-groupes cycliques, I, II, Proc. Japan Acad. 32 (1956), 736-743. [12] Z. Janko, Finite 2-groups with exactly one nonmetacyclic maximal subgroup, Israel J. Math. 166 (2008), 313-347. [13] G. A. Jones, Regular embeddings of complete bipartite graphs: classification and enumeration, Proc. London Math. Soc. (3) 101 (2010), 427-453. [14] G. A. Jones, R. Nedela, and M. Skoviera, Regular embeddings of Kn,n where n is an odd prime power, European J. Combin. 28 (2007), 1863-1675. [15] G. A. Jones, R. Nedela, and M. Skoviera, Complete bipartite graphs with a unique regular embedding, J. Combin. Theory Ser. B 98 (2008), 241-248. [16] J. H. Kwak and Y. S. Kwon, Regular orientable embeddings of complete bipartite graphs, J. Graph Theory 50 (2005), 105-122. [17] J. H. Kwak and Y. S. Kwon, Classification of reflexible regular embeddings and self-Petrie dual regular embeddings of complete bipartite graphs, Discrete Math. 308 (2008), 2156-2166. [18] R. Nedela, M. Skoviera, and A. Zlatos, Regular embeddings of complete bipartite graphs, Discrete Math 258 (1-3) (2002), 379-381. [19] H. Wielandt, Über das Produkt von paarweise vertauschbaren nilpotenten Gruppen, Math. Z. 55 (1951), 1-7. d MFA ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 6 (2013) 171-185 Sharp spectral inequalities for connected bipartite graphs with maximal Q--index Milica AndeliC * Department of Mathematics, University ofAveiro, 3810-193 Aveiro, Portugal Faculty of Mathematics, University of Belgrade, 11 000 Belgrade, Serbia C. M. da Fonsecaf Department of Mathematics, University of Coimbra, 3001-501 Coimbra, Portugal Tamara Koledin Faculty of Electrical Engineering, University of Belgrade, 11 000 Belgrade, Serbia Zoran Stanic * Faculty of Mathematics, University of Belgrade, 11 000 Belgrade, Serbia Received 31 October 2011, accepted 26 December 2011, published online 11 July 2012 Abstract The Q-index of a simple graph is the largest eigenvalue of its signless Laplacian. As for the adjacency spectrum, we will show that in the set of connected bipartite graphs with fixed order and size, the bipartite graphs with maximal Q-index are the double nested graphs. We provide a sequence of (in)equalities regarding the principal eigenvector of the signless Laplacian of double nested graphs and apply these results to obtain some lower and upper bounds for their Q-index. In the end, we give some computational results in order to compare these bounds. Keywords: Double nested graph, signless Laplacian, largest eigenvalue, spectral inequalities. Math. Subj. Class.: 05C50 * Research supported by CIDMA - Center for Research and Development in Mathematics and Applications, FCT - Funda^ao para a Ciencia e a Tecnologia, through European program COMPETE/FEDER and Serbian Ministry of Education and Science, Project 174033. t Research supported by CMUC - Centro de Matematica da Universidade de Coimbra and FCT - Funda^ao para a Ciencia e a Tecnologia, through European program COMPETE/FEDER * Research supported by Serbian Ministry of Education and Science, Projects 174012 and 174033. E-mail addresses: milica.andelic@ua.pt (Milica Andelic), cmf@mat.uc.pt (C. M. da Fonseca), tamara@etf.rs (Tamara Koledin), zstanic@math.rs (Zoran Stanic) 1 Introduction Let G = (V, E) be a (simple) graph, of order v = |V| and size e = |EThe signless Laplacian of G is defined to be the matrix Q = A + D, where A(= A(G)) is the adjacency matrix of G, while D(= D(G)) is the diagonal matrix of its vertex degrees. The largest eigenvalue (or spectral radius) of Q is usually called the Q-index of G, and is denoted by k(= k(G)). Much interest has been paid recently to this very important spectral invariant. Let us recall that A + 1 < k < 2(v - 1), (1.1) where A denotes the maximal vertex degree of the graph, with equality for stars, on the lower bound, and complete graphs, on upper bound [7]. In 2007, Cvetkovic, Rowlinson, and Simic [6] conjectured that k < v - 1 + d, (1.2) where d is the average (vertex) degree of a graph. Later Feng and Yu [9] proved that (1.2) is true (cf. also [1]). Many other bounds on Q-index for arbitrary (connected) graphs can be found in [8]. We will now describe in brief the structure of a connected double nested graph (or DNG for short). It was first considered in [3, 4] and, independently, under the name of chain graph, in [5], in studying graphs whose least eigenvalue is minimal among the connected (bipartite) graphs of fixed order and size. The vertex set of any such graph G consists of two colour classes (or co-cliques). To specify the nesting, both of them are partitioned h h into h non-empty cells Ui and Vi, respectively; all vertices in Us are joined (by i=1 i=1 h+1-s cross edges) to all vertices in Vk, for s = 1, 2,..., h. Denote by NG(w) the set of k = 1 neighbors of a vertex w. Hence, if u' G Us+1, u'' G Us, v' G Vt+1, and v'' G Vt, then NG(u') C NG(u'') and NG(v') C NG(v''). This claim makes precise the double nesting property. Observe that 1 < s,t < h. If ms = |Us | and ns = |Vs |, with s = 1, 2,..., h, then G is denoted by DNG(m 1, m2,..., mh; U1,n2,..., nh) . Note that G is connected whenever m1, n1 > 0. Additionally, if some of the remaining parameters are equal to zero, we again get a DNG with a smaller value of h. Thus, throughout we assume that all these parameters are greater than zero. We now introduce some notation to be used later on. Let s t Ms = ^^ mi and Nt = ^^ nj , for 1 < s, t < h. i=1 j=1 Thus G is of order v = Mh + Nh and size e = J2 h=1 ms Nh+1-s. Observe that Nh+1-s is the degree of a vertex u g Us; the degree of a vertex v g Vt is equal to Mh+1_t. We also set Ms,t = Mt - Ms-1 and, additionally, MM = Mt. Figure 1: The structure of a double nested graph. 2 Extremal bipartite graphs Let G be a bipartite graph with colour classes U and V. First, we state the main result of this section. Theorem 2.1. If G is a graph for which k(G) is maximal among all connected bipartite graphs of order v and size e, then G is a DNG with all pendant edges attached to a common vertex. Theorem 2.1 means that double nested graphs play the same role among bipartite graphs (with respect to the signless Laplacian index) as nested split graphs among non-bipartite graphs. The same classes of graphs appear as extremal with respect to the adjacency spectra as well, i.e., in the class of all connected (resp. all connected bipartite) graphs of fixed order and size, those with maximal radius with respect to the adjacency matrix are NSGs (resp. DNGs). The proof of Theorem 2.1 is based on the following lemmas, the first of which is taken from [6]. Recall that there exists a unique unit eigenvector corresponding to k(G) having only positive entries; this eigenvector is called the principal eigenvector of G. Lemma 2.2. Let G' be the graph obtained from a connected graph G by rotating the edge rs around r to the non-edge position rt. Let x = (x1,x2,... ,xv )T be the principal eigenvector of G. If xt > xs then k(G') > k(G). The next lemma will be very helpful when we find a bridge in a graph whose index is assumed to be maximal. Given two rooted graphs P(= Pu) and Q(= Qv) with u and v as roots, let G be the graph obtained from the disjoint union P Ù Q by adding the edge uv. Let G' be the graph obtained from the coalescence of Pu and Qv by attaching a pendant edge at the vertex identified with u and v. Lemma 2.3. With the above notation, if P and Q are two non-trivial connected graphs then k(G) < k(G'). Proof. Let (x1, x2,..., xv )T be the principal eigenvector of G. Without loss of generality, we may suppose that xu ^ xv. Let Np (u) be the neigbourhood of u in P ; since P is non-trivial, NP(u) = 0. Now G' is obtained from G by replacing the edges uw, with w G NP(u) by the edges vw and, therefore, k(G) < k(G'), by Lemma 2.2, as required. □ In what follows we assume that G has maximal index among the connected bipartite graphs of fixed order and size. Lemma 2.4. Let G be a graph satisfying the above assumptions, andlet x = (#i,... ,xv ) be the principal eigenvector of G. If v and w are vertices in the same colour class such that xv > xw, then deg(v) > deg(w). Proof. Let U, V be the colour classes of G. Assuming that v and w are vertices in V such that xv > xw and deg(v) < deg(w), then deg(w) > 1 and there exists u G U such that v u ~ w. By Lemma 2.3, we may rotate uw to uv to obtain a graph G' such that k(G') > k(G). If uw is a bridge, then deg(u) = 1 and, again by Lemma 2.3, G' is necessarily connected; but now the maximality of k(G) is contradicted and the proof is complete. □ From now on we take the colour classes to be U = {u1,u2,... ,um} and V = {vi,v2,...,v„}, with xMl > x„2 > ••• > x„m and xvi > xv2 > ••• > xv„. By Lemma 2.4, this ordering coincides with the ordering by degrees in each colour class. In the next lemma we note some consequences of those facts. Lemma 2.5. Let G be a graph satisfying the above assumptions including those on vertex ordering. Then (i) the vertices ui and vi are adjacent; (ii) u1 is adjacent to every vertex in V, and v1 is adjacent to every vertex in U ; (iii) if the vertex u is adjacent to vk, then u is adjacent to vj, for all j < k, and if the vertex v is adjacent to uk, then v is adjacent to uj, for all j < k. Proof. First we consider bridges in G. By Lemma 2.3, all bridges are pendant edges. By Lemma 2.2, all pendant edges are attached at the same vertex, and this vertex w is such that xw is maximal. Without loss of generality, xui > xvi and w = u1. It follows that the result holds if G is a tree and, consequently, G is a star. Accordingly, we suppose that G is not a tree. To prove (i), suppose by way of contradiction that u1 v1. Then v1 is adjacent to some vertex u G U, and uv1 is not a bridge. By Lemma 2.2, we may rotate v1u to v1u1 to obtain a connected bipartite graph G' such that k(G') > k(G), contradicting the maximality of K(G). To prove (ii), suppose that u is a vertex of U not adjacent to v1. Then u = u1 by (i), uv is not a bridge, and u is adjacent to some vertex v in V other than v1. Now we can rotate uv to uv1 to obtain a contradiction as before. Secondly, suppose that v is a vertex of V not adjacent to u1. Then v = v1 by (i), again vu1 is not a bridge, and a rotation about v yields a contradiction. To prove (iii), suppose that u G U, u ~ vk and u ^ v j for some j < k. Now u = u1 by (ii), and so uvk is not a bridge. Then we can rotate uvk to uvj to obtain a contradiction. Finally, suppose that v G V, v ~ vk and v uj for some j < k. In this case, vuk is not a bridge because k > 1, and the rotation of vuk to vuj yields a contradiction. The proof is now finished. □ Taking into account Lemma 2.5 and the definition of a DNG the first part of Theorem 2.1 follows. It remains only to prove that all cut-edges in the observed DNG are pendant edges attached to a common vertex. This easily comes from Lemma 2.3. 3 Q-eigenvectors of DNGs Here we consider the principal eigenvector of the signless Laplacian of DNGs. In this section (and in the next one, if not told otherwise) we will assume that ^C — ( ^X1, . . . , ^X v ) is a Q-eigenvector of G with all positive entries, which is usually normalized, i.e., v = i. i=1 The entries of x are also called the weights of the corresponding vertices. We first observe that all vertices within the sets Us or Vt, for 1 < s,t < h, have the same weights, since they belong to the same orbit of G. Let X* u — üg, if u G Us, while xv = bt, if v G Vt. From the eigenvalue equations for k, applied to any vertex from Us or Vt, we get h+1-s Küs = Nh+1-süs + nj bj , for s = 1,...,h, (3.1) j=1 and h+1-t Kb t = Mh+1-tbt + miüi, for t = 1,..., h. (3.2) i= 1 By normalization we have and, from (3.1), we easily get hh miai + nj bj = 1, i=1 j=1 (3.4) (3.5) for s = 1,...,h, (3.6) 1 h+1-s -—- > n, bo , for s = 1,..., h. K - Nh+1-s ... 3=1 From (3.2) we have 1 h+1-t bt =-77- y] miai, for t = 1,..., h, K - Mh+1-t ^ - i=1 and therefore, using (3.3), we have 1 i h h K - Nh^ l1 - £ miüi - E njbj * i=1 j=h+2-s üs= üs = or, using (3.2) for t = 1, if h \ 1 — (k — Mh)b1 — ^^ nj bj I , for s = 1,...,h. K — Nh+1-s v . , , „ » j=h+2-s Similarly, 1 — (k — Nh)a1 — miai] , for t = 1,... ,h. K — Mh + 1 _t \ h+1 t V i=h+2-t / Setting ah+1 = bh+1 = 0 and N0 = 0, from (3.4) and (3.6), together with (3.3), we get successively (k — Nh-s)as+1 — (k — Nh+i-s)as = —nh+i-sbh+i-s, for s = 1,..., h — 1, and (k — n{)ah = n1b1, for s = h. Since all components of x are positive and k > A + 1 (1.1), it comes as+1 < as, for s = 1,..., h — 1, (3.7) and bt+1 < bt, for t = 1,..., h — 1. (3.8) Furthermore, by setting s = h in (3.2), we obtain (k — m1)bh = m1a1. (3.9) Moreover, substituting s = 1 in (3.1) and t = 1 in (3.2) and applying in (3.3) we get (k — Nh)a1 + (k — Mh)b1 = 1, and finally as = -r1- ( (k — Nh)a1 — ]T nj bj ) . (3.10) K — Nh+1-s \ j=h+2-s J Next we focus our attention on bounding ai's and bj's. Lemma 3.1. For any s = 1,..., h, we have Nh+1-sbh+1-s , , Nh+1-sb1 -^- < as < -^- . (3.11) K — Nh+1-s K — Nh+1-s Proof. From (3.4), we have 1 h+1-s as = - > n,-b,- . s K — Nh+1-s j=1 j j J=1 Therefore, (3.11) immediately follows since bj's are strictly decreasing, from (3.8). □ 1 Lemma 3.2. For any s = 1,... ,h, , f, Nh+2-s,h{* mi \\ „ as < a i 1---—:— 1 H--, (3.12) V k - Nh+i-s V K - mi J J Proof. The inequality (3.12) follows from (3.10), since bj's are strictly decreasing, bearing in mind (3.9) as well. □ Lemma 3.3. For any s = 1,... ,h, i ai , s- nh+i-iMi a- > ^^ 1 "E K - Nh+i-s \ , K - Mi \ i=i / Proof. By induction on s. For s = 1, the inequality holds trivially. Assume next that -i ai nh+i-iM, a' > ^^ 1 -E K - Nh+i-s \ , K - Mi v i=i for s > 1. Then as+i 1 h- K - Nh-s 1 K - Nh-s ai K - Nh-s ai j=i ((k - Nh+ i-s )as - Nh+ i-s bh+ i-s ) s-1 1- nh+i-iMA Nh+i-sMsai i= i K - Mi (K - Nh-s)(K - Ms) ^ nh+i-iMA K - Nh-s \ = K - Mi J . This ends the proof. □ Lemma 3.4. For any s = 1,..., h, we have as < ^- (Nh+i-s - ( Kf\+i-s M ) Ì , (3.13) K - Nh+i-s \ (k - ni)(K - Ms), where h+i-s fh+i-s = njMh+2-j,h ■ j=i Proof. From (3.4) and (3.12) applied to bj, we get 1 h+i-s E njb- jj K - Nh+i-s . -, j=i ^ -s1-hys njbi( 1 - Mh+-j,h (1 + ^- K - Nh+i-s \ K - Mh+ i-j V K - ni bi Kfh+i — s ^ -b- I Nh+i-s - fh+i K - Nh+i-s \ s (k - ni)(K - Ms) J The proof is now complete. □ 4 Some bounds on the Q-index of a DNG In this section we obtain some upper and lower bounds on the Q -index of DNGs using the eigenvalue and the matrix technique. We also emphasize that our main goal is to consider the estimation of the Q-index of large graphs. Before we proceed, we provide the following observations. First, if h =1 we get a complete bipartite graph Kmi,ni, whose Q-index is equal to m1 + n1 = v [2]. Furthermore, since the Q-index of an arbitrary graph increases by inserting edges (cf. [6]), we have K < v, (4.1) for any (not necessarily connected) DNG. Otherwise, if h > 1 is fixed but the graph size is not, using the same previous arguments, the maximal Q-index would appear in DNG(m1, 1,..., 1; n1,1,..., 1). The computational results suggest this will happen when |m1 - n11 < 1. So, these cases are not interesting for our research and, therefore, we will assume that h > 1 and the size is fixed. 4.1 Eigenvalue technique Now we establish some bounds for the Q-index of DNGs using the eigenvalue technique. We start with lower bounds. Proposition 4.1. If G is a connected DNG, then k > max {Mh+1-k + Nk}. Proof. On the one hand, from (3.2), we get 1 h+1-k , 1 V"1 . Mh+1-k «h+1-k bk = -7"7- V miai > -^- , K - Mh+1-k = K - Mh+1-k since, from (3.7), Oj's are decreasing. On the other hand, from (3.1), we get 1 ^ , . Nk bk «h+1-k =-rf b1 ^ -\T , + k - Nk -f-f j j k - Nk 1=1 since bj's are decreasing, from (3.8). From the last two inequalities we get K(K - (Mh+1-k + Nk)) > 0, which is equivalent to K > Mh+1-k + Nk. In particular, for k = h and k = 1, we obtain the following corollary. Corollary 4.2. If G is a connected DNG, then k > m1 + Nh and k > n1 + Mh . □ Proposition 4.3. If G is a connected DNG, then k » 2 (f+i+v ('- i)2+4eh where t = Eh=i miNh+i-i and eh = V m4. Vh mN h ^ Nh i miNh+i-i i= 1 h Proof. Let y = (y1,... ,yv )T be a vector whose components are indexed by the vertices of G, and let yu = Nh+1-i if u e Ui, for some i e {1,..., h}, or, otherwise, yv = q = k -1, for some t, if v e Vj for some j e {1,... ,h}. Substituting y into the Rayleigh quotient (see, e.g., [8, p. 49]) we obtain ^ 2 Eh=i miNh-1+iq + Eh=i mNj+i-i + Eh= 1 niMh-1+iq2 K » -!-r-!- Eh=1 miNh+1-i + Nhq2 yT Qy due to Rayleigh's principle which reads —^— < k. Since q = k -1, we get yT y h h h Nhq3 + (Nht - e)q2 - £ miN^-A » £ m^N^-i - t £ miNh+1-i. i=1 i=1 i=1 Choosing t E,h=1 miNh+1-i Ei=1 miNh+1-i and having in mind that N1 < t < Nh, we immediately get a quadratic inequality in q and the proof is concluded. □ Proposition 4.4. If G is a connected DNG, then K < 2 (v + Vv2 - 4(MhNh - e)) . (4.2) Proof. From (3.1), with s = h, and from (3.3), recalling (3.11), we get h h n (k - Mh)b1 = £ mi ai < £ mi n N+ —— . i=1 i=1 K h+1-s Then, we obtain (k - Mh)(K - Nh) < e, and, therefore, from the quadratic inequality k2 - (Mh + Nh)K + MhNh - e < 0, we obtain k1 < k < k2 where k1 and k2 are the solutions of the associated quadratic equality, and this completes the proof. □ The next two bounds improve the bound (4.2). We recall that fh+1-i is defined in Lemma 3.4. Proposition 4.5. If G is a connected DNG, then - < 2 (v + V v2 - 4(MhNh - e')) , where ' v(v - Nh) , e = e - -,--r y. miJh+1-i. (v — ni)2(v — mi) z—' i=1 Proof. As in the proof of Proposition 4.4, we have h (- - Mh)b 1 = ^2 miai. i= 1 Using (3.12), we get k - Mh < £ -^h- fNh+1-i - ( fT- M ) ) , = - - Nh+ 1-i V (- - n1)(- - Mi) J and therefore (- - Mh)(- - Nh) < e - ---(-,2/Nh)-r £ mifh+1-i . (- — n1)2(- — m1) i=1 Taking into account that - < v, from Proposition 4.4 it follows (- - Mh)(- - Nh) < e', and the proof ends. □ The next result may be proved in a similar way. Proposition 4.6. If G is a connected DNG, then - < 2 („ + Vv2 - 4(MhNh - e'')) , where « -'(-' - Nh) A f e = e - (-' - m)2(-' - m!^ mifh+1-i, i=1 for -' =2 (v + Vv2 - 4(MhNh - e')) . 4.2 Matrix technique The partition hh V = U Uk U U Vk (4.3) k=1 k=1 is equitable since every vertex in Ui and every vertex in Vi have the same number of neighbors in Uj and Vj, for all i, j e {1,2,..., h}. Let AD be the signless Laplacian divisor matrix of a DNG(m1,..., mh; n1, ..., nh) with respect to the equitable partition (4.3). The matrix AD has the following form: Nh An = Nh 1 N2 N1 m1 m1 m1 m1 m2 m2 m2 mh-1 mh mh 1 n1 n1 n1 n1 n2 n2 n2 nh-1 nh 1 nh Mh Mh 1 M2 M1 where the non-mentioned entries are to be read as zero. Setting N= n1 n2 n1 n2 n1 n2 n1 nh-1 nh nh 1 \ / M= m1 m2 m1 m2 m1 m2 m1 mh-1 mh mh 1 / D1 = diag(Nh,..., N1), and D2 = diag(Mh,..., M1), AD can be rewritten in the compact block form D1 N M Do An In order to obtain more bounds we set P = for some x = 0. Since the matrices PAd P-1 0 xi 1 0 D2 -1N xM D1 and Ad are similar, they have the same index. We choose x such that the sum in the first row and the (h + 1)-th row are equal. It leads to Mhx2 — (Nh — Mh)x — Nh = 0, i.e., x = M. By Frobenius Theorem [11, Theorem 3.1.1], we have min Ri ^ k ^ max Ri, 1 v (compare (4.1)), so next we use the result of Minc (see [10]), which for the matrix AD reads: • ^j = 1(AD)ijRj' . Xj = 1(AD)ijRj min —--—- ^ k ^ max —--—-. i Ri i Ri Proposition 4.8. If G is a connected DNG, then min{n1 + Mh, m-1 + Nh} < k < max j N- + Nh, M + Mh j . (4.6) The bounds (4.6) obviously improve both (4.4) and (4.5), but the lower bound is still rough comparing with Corollary 4.2. 5 Computational results In this final section, we provide several examples which can help to gain a better insight into the quality of the bounds obtained in the previous section. We compute the lower bounds of Propositions 4.1 and 4.3, and the upper bounds from Propositions 4.4, 4.5, 4.6, and 4.8. One observes that the bound from Proposition 4.1 is always integral. The number of vertices in the corresponding DNG is also given in every example since it makes another upper bound (cf. (4.1)). It can be easy checked that the lower bounds from Corollary 4.2 and Propositions 4.7 and 4.8, all of them having simple expressions, are rough in some cases, and therefore they are not considered in our examples. We also compute the relative errors in each case. Example 5.1. First we consider a randomly chosen DNG with small number of vertices and some larger DNGs derived from the previous one: G1 = DNG(2, 2, 5, 3; 2, 3,1,1); G2 = DNG(10, 10, 25,15; 10,15, 5, 5); G3 = DNG(200, 200, 500, 300; 200, 300,100,100); Prop. 4.3 Prop. 4.1 K Prop. 4.8 Prop 4.6 Prop 4.5 Prop 4.4 V Gi 13.6785 -12.57% 14 -10.52% 15.6451 16.7500 7.06% 17.0210 8.79% 17.0550 9.01% 17.4530 11.56% 19 21.44% G2 68.3923 -12.57% 70 -10.52% 78.2257 83.7500 7.06% 85.1052 8.79% 85.2749 9.01% 87.2649 11.56% 95 21.44% G3 1367.8452 -12.57% 1400 -10.52% 1564.5133 1675.0000 7.06% 1702.1030 8.79% 1705.4985 9.01% 1745.2987 11.56% 1900 21.44% Notice that k(G2) (resp. k(G3)) is very close to 5k(Gi) (resp. 100k(G3)); we get 5k(Gi) - k(G2) « l0-7. Since the similar fact holds for all bounds obtained (compare the corresponding propositions), we get the same results for the relative errors. Example 5.2. Here we consider the DNGs obtained from Gi by multiplying some of its parameters: H1 = DNG(2000, 2, 5, 3; 2, 3,1,1000); H2 = DNg(2000, 2, 5, 3; 2, 3,1000, l); H3 = dng(2000, 2, 5, 3; 2, 3000, l, l); H4 = dng(2000, 2, 5, 3; 2000, 3, l, l); Prop. 4.3 Prop. 4.1 K Prop. 4.8 Prop 4.6 Prop 4.5 Prop 4.4 V Hi 3006.0284 -8 • 10-6% 3006 -1 • 10-3% 3006.0287 3011.0164 0.17% 3008.2682 0.07% 3008.2960 0.08% 3012.6750 0.22% 3016 0.33% H2 3008.0175 -5 • 10-6% 3007 -0.03% 3008.0177 3012.0104 0.13% 3009.8145 0.06% 3009.8323 0.06% 3013.3388 0.18% 3016 0.27% H3 5010.9908 -5 • 10-7% 5009 -0.04% 5010.9909 5010.9980 1 • 10-4% 5011.7199 0.15% 5011.7199 0.15% 5012.2008 0.02% 5014 0.06% Hi 4014.9731 -3 • 10-6% 4010 -0.12% 4014.9732 4014.9866 3 • 10-4% 4014.9800 2 • 10-4% 4014.9800 2 • 10-4% 4014.9933 5 • 10-4% 4015 7 • 10-4% In this example all bounds are (more or less) close to the exact value of Q-index. We already pointed that the bounds obtained in Propositions 4.5 and 4.6 are the improvements of the one obtained in Proposition 4.4. This example shows that the bound from Proposition 4.8 is incomparable to them. In opposition to the previous example, here Proposition 4.1 gives a better estimation than Proposition 4.3. Example 5.3. The parameters of the following DNGs are obtained by multiplying the parameters of G1 by l, 10,100 or 1000 ad hoc. 11 = DNG(2, 2, 5, 3; 2000, 300,10, l); 12 = dng(2, 2, 5, 3; 2, 30,100, l000); 13 = dng(2000, 200, 50, 30; 2000, 300,10, l); Prop. 4.3 Prop. 4.1 K Prop. 4.8 Prop 4.6 Prop 4.5 Prop 4.4 V Ii 2255.0867 -2.65% 2314 -0.10% 2316.3632 2322.5716 0.26% 2322.5716 0.27% 2322.5733 0.27% 2322.5737 0.27% 2323 0.29% 12 1118.5026 -1.37% 1134 -6 • 10-5% 1134.0007 1134.3799 0.03% 1134.4002 0.04% 1134.4000 0.04% 1134.4002 0.04% 1144 0.88% I3 4562.6064 -0.28% 4550 -1 • 10-3% 4562.6584 4563.2717 0.01% 4563.1367 0.01% 4563.1369 0.01% 4563.6312 0.02% 4591 0.62% Taking into account the lower bound of Proposition 4.3, one can conclude that its deviation from the exact value is expected for Ii (and other similar graphs). Note that Proposition 4.6 will often give better bound than Proposition 4.5, but not always - see graphs I2 or J2 in the next example. Example 5.4. Finally, we consider the extensions of the original graphs: Ji = DNG(2, 2, 5, 3, 2, 3,1,1; 2, 3,1,1, 2, 2, 5, 3); J2 = DNG(20000, 2, 5, 3,10,10,10,10; 2, 3,1,10000,10,10,10,10); J3 = DNG(2, 2, 5, 3,1,1,1,1; 2000, 300,10,1,1,1,1,1); Prop. 4.3 Prop. 4.1 K Prop. 4.8 Prop 4.6 Prop 4.5 Prop 4.4 V Jl 23.1888 -15.56% 23 -16.24% 27.4601 29.0526 5.80% 32.3032 17.64% 32.3022 17.64% 32.8203 19.52% 38 38.38% J2 30065.9176 -6 • 10-7% 30046 -0.07% 30065.9178 30080.9446 0.05% 30072.6747 0.02% 30072.7003 0.02% 30085.9668 0.07% 30096 0.10% J3 2313.0140 -0.62% 2324 -0.15% 2327.5409 2330.8445 0.14% 2330.8452 0.14% 2330.8452 0.14% 2330.8456 0.14% 2331 0.15% The bounds obtained will be used in a forthcoming research regarding the graphs with maximal Q-index and fixed (but high) orders and also a fixed particular size. We also remark that the results could be also compared to the corresponding bounds obtained for the adjacency spectra. Acknowledgment The authors thank two anonymous referees for providing constructive comments in improving the contents of this paper. References [1] M. Andelic, C. M. da Fonseca, S. K. Simic and D. V. Tošic, Connected graphs of fixed order and size with maximal Q-index: Some spectral bounds, Discrete Appl. Math. 160 (2012), 448-459. [2] M. Anđelic, T. Koledin and Z. Stanic, Nested graphs with bounded second largest (signless Laplacian) eigenvalue, Electron. J. Linear Algebra 24 (2012), to appear. [3] F. K. Bell, D. Cvetkovic, P. Rowlinson and S. K. Simic, Graphs for which the least eigenvalue is minimal, I, Linear Algebra Appl. 429 (2008), 234-241. [4] F. K. Bell, D. Cvetkovic, P. Rowlinson and S. K. Simic, Graphs for which the least eigenvalue is minimal, II, Linear Algebra Appl. 429 (2008), 2168-2179. [5] A. Bhattacharya, S. Friedland and U. N. Peled, On the first eigenvalue of bipartite graphs, Electron. J. Combin. 15 (2008), R#144. [6] D. Cvetkovic, P. Rowlinson and S. K. Simic, Eigenvalue bounds for the signless Laplacian, Publ. Inst. Math. (Beograd), 81 (95) (2007), 11-27. [7] D. Cvetkovic, P. Rowlinson and S. K. Simic, Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007), 155-171. [8] D. Cvetkovic, P. Rowlinson and S. K. Simic, An Introduction to the Theory of Graph Spectra, Cambridge University Press, 2009. [9] L.-H. Feng, G.-H. Yu, On three conjectures involving the signless Laplacian spectral radius of graphs, Publ. Inst. Math. (Beograd) 85 (99) (2009), 35-38. [10] S.-L. Liu, Bounds for the greatest characteristic root of a nonnegative matrix, Linear Algebra Appl. 239 (1996), 151-160. [11] M. Marcus and H. Minc, A Survey of Matrix Theory and Matrix Inequalities, Dover Publications Inc., New York, 1992; unabridged reprint of the corrected 1969 printing, Prindle, Weber, & Schmidt, Boston. ARS MATHEMATICA CONTEMPORANEA Author Guidelines Papers should be prepared in LTEX and submitted as a PDF file. 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After a short period as an Assistant Professor at the University of Minnesota, he joined the faculty of Mathematics at the Ohio-State University, where he worked for the rest of his professional career. Thanks to his outgoing personality, he had from the beginning a diverse group of collaborators, which led him to tackle problems in Fixed Point Theory and questions on Graphs. When during the early 1970's localization techniques in topology were developed, he made immediate use of the new tool to obtain interesting results on vector fields on manifolds. At the same time, he advanced his work on graph theory, culminating in a classification of the irreducible graphs for the projective plane. After about 1980, he began a productive collaboration on questions in geometric group theory, dealing with group cohomology involving the mapping class group as well as the automorphism group of a free group. 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ARS MATHEMATICA CONTEMPORANEA UP IAM - University of Primorska Muzejski trg 2, 6000 Koper, Slovenia ANNOUNCEMENT PhD Fellowship ("Young Researcher" position) at the University of Primorska, Slovenia The University of Primorska announces three "Young researcher" positions under the supervision of • Dragan MarusiC (Algebra and Discrete Mathematics); • Enes Pasalic (Cryptography); • Dragan Stevanovic (Spectral Graph Theory, Applications). Applicants should have a BSc or equivalent training in Mathematics (by September 30, 2013), and are expected to enroll in the PhD program. The positions are for 3 and 1/2 years and include a tuition fee waiver. The holder is expected to complete his/her PhD training in 4 years. The deadline for applications is July 31, 2013. 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