YET ANOTHER LOOK AT THE GRAY GRAPH. Tomaž Pisanski 1. In loving memory of my parents

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 36 (2007), YET ANOTHER LOOK AT THE GRAY GRAPH Tomaž Pisanski 1 (Received November 2006) In loving memory of my parents Abstract The Gray graph is the smallest trivalent semisymmetric graph; it has 54 vertices We present three descriptions of this remarkable graph and establish their equivalence The construction readily generalizes in order to produce a series of semisymmetric graphs of arbitrary valence n > 2 Also, this implies the existence of geometric triangle-free point-transitive, line-transitive, flag-transitive non self-dual configurations of type (n n n), for any n > 2 These graphs have been studied already by R Foster and I Bouwer The Gray graph appears as the medial layer graph of a regular 4-polytope The combinatorial structure of this polytope is explained in detail A relationship between the Gray graph and the Pappus graph is presented The Gray graph can be constructed by taking three copies of the complete bipartite graph K 3,3 and, for each edge e of K 3,3, subdividing e in each of the three copies, and joining the three resulting vertices of valence 2 to a new vertex; see for instance [4] It has been shown that the Gray graph is the unique smallest trivalent semi-symmetric graph; see [19] In particular, this means that any trivalent edgebut not vertex-transitive graph on 54 vertices is isomorphic to the Gray graph Let H 1 H 2 G be graphs Define G(H 1, H 2 ) to be a bipartite graph defined on all occurrences of H 1 as an induced subgraph in G and all occurrences of H 2 as an induced subgraph of G and joining an occurrence of H 1 to an occurrence of H 2 if and only if the occurrence of H 1 is an induced subgraph of the corresponding occurrence of H 2 The graph G(H 1, H 2 ) will be called the Grassmannian of G with respect to H 1, H 2 The name is chosen by analogy with Grassmannians traditionally defined on the collection of k-dimensional vector subspaces of an n-dimensional vector space For a given graph G, let G (2) denote the graph on the same vertex set in which two vertices are adjacent if and only if they are at distance 2 in G Let us first present constructions of three graphs B 1, B 2 and B 3 The constructions were inspired by Bouwer [6, 8]: Construction 1 Let us consider three disjoint sets on 9 elements: X = {x 1, x 2, x 3 }, Y = {y 1, y 2, y 3 }, Z = {z 1, z 2, z 3 } and their union V = X Y Z Let E denote the set of 27 pairs {x i, y j }, {x i, z k }, {y j, z k } and let T denote the set of 27 triples {x i, y j, z k } Define the bipartite graph B 1 with bipartition (E, T) in which a pair e E is adjacent to the triple t T if and only if e t 1 Supported in part by the Public Agency of Research and Development of Slovenia, Grants P1-0294,J1-6062,L1-7230

2 86 TOMAŽ PISANSKI1 Construction 2 Consider the complete tripartite graph K 3,3,3 Let B 2 be the graph K 3,3,3 (K 2, K 3 ) Figure 1 Spatial version of the Gray configuration consists of 27 points and 27 lines grouped in three classes, each containing 9 parallel lines Construction 3 Consider the cartesian product K 3 K 3 K 3 Define B 3 to be the graph K 3 K 3 K 3 (K 1, K 3 ) We may shorten the notation here: K 3,3,3 = K 3(3) and K 3 K 3 K 3 = K 3 3 Later we also denote by S(G) the subdivision graph of a given graph G, in which each edge of G is replaced by a path of length two Theorem 1 The graphs B 1, B 2, B 3 are all isomorphic to the Gray graph Proof To see that B 2 is isomorphic to B 1, label the vertices in each color class by labels from X, Y, and Z, respectively The 27 edges of K 3,3,3 are naturally labeled by labels from E Furthermore, the 27 cycles of length 3 are naturally labeled by T and the incidences of B 1 correspond precisely to the incidences in B 2 Consider the vertex set of K 3 K 3 K 3 to be X Y Z The labeling can be carried on to B 3 The vertices of B 3 can be equivalently labeled as sets {x i, y j, z k } rather than triples (x i, y j, z k ) Furthermore a triangle in K 3 K 3 K 3 is determined by two coordinates: (x i, y j, ), (x i,, z k ), (, y j, z k ), or, equivalently by an unordered pair {x i, y j }, {x i, z k }, {y j, z k } This establishes the isomorphism between B 1 and B 3 where a pair e E is mapped to a triangle of B 3 and a triple t T is mapped to a vertex of B 3 In [22] it is shown that the Gray graph is the Levi graph of the Gray configuration, the configuration of 27 points and 27 lines as depicted in Figure 1 This identifies B 3 as the Gray graph An alternative argument that the three graphs B i, i = 1, 2, 3 are isomorphic to the Gray graph follows from the fact that the graphs are trivalent and semi-symmetric on 54 vertices; see [19] They are clearly trivalent The construction of B 1 shows that the graph is edge-transitive The graph B (2) i consists of two connected components One is isomorphic to K 3 K 3 K 3 in which each edge belongs to exactly one triangle, while the other one does not have this property Hence B i is not vertex-transitive The proof of our theorem casts a new light to the Menger graph M and the dual Menger graph D of the Gray configuration Graph M, isomorphic to K 3 K 3 K 3 is defined on the vertex set T with triples t, s adjacent if and only if s t = 2

3 YET ANOTHER LOOK AT THE GRAY GRAPH 87 The graph D is defined on the set E with two pairs e, f adjacent if and only if e f = 1; see Figure 7 in [22] Compare also [8] Figure 2 The generalized Gray configuration (256 4) It is composed of four clearly visible (64 3,48 4) subconfigurations There is a generalization of our argument from the case n = 3 to general n Instead of triples, consider n-tuples Let V (n) be a set {x ij i, j Z n } of n 2 elements, and let T(n) be the set of all n-tuples whose i th entry is x ij for some j, for all i Z n Similarly, let E(n) denote the set of n n different (n 1)-tuples that are obtained from some n-tuple in T(n) by deleting any one of its entries Note that each n-tuple from T(n) gives rise to n distinct (n 1)-tuples from E(n), with each (n 1)-tuple obtained n times Then define a graph B 1 (n) on 2n n vertices or, equivalently, a triangle-free (n n n) configuration that is both combinatorial and geometric, by taking V (n) E(n) as vertex-set and letting an n-tuple v T(n) be adjacent to an (n 1)-tuple u E(n) if and only if u can be obtained from v by deleting an entry from v Figure 1 shows how to construct this geometric configuration in the case n = 3 In addition, our theorem generalizes If we define B 2 (n) = K n(n) (K n 1, K n ) and B 3 (n) = K n n(k 1, K n ), then the graphs B 1 (n), B 2 (n) and B 3 (n) are isomorphic This generalization implies the existence of geometric triangle-free point- lineand flag-transitive non self-dual configurations (v k ) for any value of k; see also [15, 16] It also implies the existence of semisymmetric graphs of arbitrarily large valence n > 2 Indeed, the graph B(n) on 2n n vertices of valence n has girth 8 and diameter 2n, and each vertex corresponding to a n-tuple has (n 1) n antipodal vertices while each vertex corresponding to a (n 1)-tuple has (n 1) n 1 antipodal vertices Marston Conder (private communication) also noticed that the two distance sequences differ at the fourth position The numbers of vertices at distances 0, 1, 2, 3 and 4 from a given n-tuple are 1, n, n(n 1), n(n 1) 2 and n(n 1) 3 /2 respectively, while the corresponding numbers from a given (n 1)-tuple are 1, n, n(n 1), n(n 1) 2 and (n 1) 4 Both arguments establishing semisymmetry of B(n) differ from the original one, given by Bouwer, where 3- or 4-arc transitivity is used for each color class of vertices The automorphism group of this graph is

4 88 TOMAŽ PISANSKI1 the wreath product S n S n, of order (n!) n+1, which acts imprimitively on the set of n-tuples, with n blocks of size n This infinite class of Bouwer graphs seems to admit simpler description than the one proposed in [12] The construction of the configurations naturally generalizes not only to (n n n ), but to configurations (p q, n k ) for all q and k (with appropriate, easily computable p and n): In q-dimensional space, take a lattice hypercube with k points on each side; see [8] By a suitable projection in the plane we obtain a geometric configuration of points and lines This observation can be written in a formal form Theorem 2 For any positive integers q and k, there exists a point-transitive, line-transitive, flag-transitive, triangle-free, geometric configuration (p q, n k ), where p = k q and n = qk q 1 The configuration is self-dual and and only if q = k = 2 The Levi graphs B(n, k) in this case admit the following equivalent descriptions: B(n, k) = K k n(k 1, K n ) = K k(n) (K n 1, K n ) These configurations and hence their Levi graphs were first considered by Bouwer and therefore deserve to be named after him The Gray graph is therefore isomorphic to B(3, 3) = B(3) K 1, S(K (9) 3 ) 2 3 Gray 3 2 S(K 3,3,3 ) 6 1 Figure 3 Link figure for the Gray polytope, a regular 4-polytope, of type {3, 6, 3} K 9, S(K (3) S(K 3,3 ) K 1, ) K 6,1 6 Figure 4 The facet of the Gray polytope is a regular 3-polytope, of type {3, 6} whose skeleton is K (3) 3 Its dual is hexagonal embedding of K 3,3 in the torus 1 18K 1, S(P appus) 2 6 S(K 3,3,3 ) 2 1 K 9,1 9 1 Figure 5 The vertex-figure of the Gray polytope is a regular 3- polytope, of type {6, 3} It is defined by a hexagonal embedding of the Pappus graph in the torus Let us conclude with an unusual role of B(3) The Gray graph arises as the medial layer graph of a certain abstract 4-polytope, as explained in [23] This polytope, that we call the Gray polytope, is depicted in Figure 3 using its link figure, as explained for other examples in [9] The flags of the Gray polytope can be viewed as elements from {1} W (X Y Z) E (X Y Z) {1}, where the set W is defined as W = {0, 1, 2} A typical flag is therefore a tuple: (1, w, (x i, y j, z k ), {p, q}, r, 1) with r {p, q} {x i, y j, z k } and w + i + j + k 0 mod 3 Since the polytope is not self-dual, we have to consider it together with its

5 YET ANOTHER LOOK AT THE GRAY GRAPH 89 dual Each vertex-figure is isomorphic to the regular map {6, 3} 1,1 and each facet is isomorphic to the regular map {3, 6} 3,0 For the dual these are the other way round Both maps and their duals are very well known One of them represents the renowned Pappus graph in the torus For a recent study of symmetric hexagonal tessellations of the torus by cubic graphs, see [21] Its dual is a regular triangulation of the torus whose skeleton is K 3,3,3 It was used, for instance, in [5] It is not hard to verify that the conditions for an abstract polytope are fulfilled In particular one can verify the diamond condition at each layer Rank 0: between 1 and (x i, y j, z k ) we have u and v where {u, v} := W \ {i + j + k mod 3} Rank 1: between w and {x i, y j } we have (x i, y j, z k ) and (x i, y j, z m ) where k and m are congruent mod 3 to the two numbers in W \ {i + j + k mod 3} Rank 2: between (x i, y j, z k ) and x i we have {x i, y j } and {x i, z k } Rank 3: between {x i, y j } and 1 we have {x i } and {y j } Furthermore, since the role of sets X, Y, Z is totally symmetric and the same is true for the subscripts in each set, one can readily verify that the polytope is flag-transitive, hence regular There are 324 automorphisms This number can be computed just by counting the number of flags Note that the link figure of a polytope represents a concise description of the ranked poset of the polytope Since the rank 4 polytope in Figure 3 is regular, it has a unique vertex figure as shown in Figure 4, and a unique facet as depicted in Figure 5 The corresponding rank 3 polytopes represent maps, which carry almostcomplete information of the Hasse diagram of the poset If the improper faces are deleted from the Hasse diagram of the facet (Figure 4) and the vertex-figure (Figure 5), then we can draw the remaining part of the corresponding Hasse diagrams on the torus as in Figures 6 and 7 Figure 6 The superposition of the hexagonal embedding of K 3,3 in the torus with its dual can be interpreted as a Hasse diagram of the facet of the Gray polytope with the two improper faces removed

6 90 TOMAŽ PISANSKI1 Figure 7 The hexagonal embedding of the Pappus graph in the torus Its dual is the triangulation of the torus by K 3,3,3 Figure 8 The vertices of the Pappus graphs are labeled by triples i, j, k that represent triples (x i, y j, z k ) Note that there are 18 triples with the property i + j + k 0 mod 3 Two triples are adjacent if and only if they agree in two coordinates Hence each of the 27 edges can be uniquely labeler by a pair (x i, y j),(x i, z k ) or (y j, z k ) inducing a natural 3-edge-coloring of the Pappus graph An unusual drawing of the Pappus graph The coordinates of each vertex is determined by the labeling given in the previous Figure Each edge is parallel to one of the three axes This gives a natural 3-edge-coloring of the Pappus graph The drawing can also be interpreted as a spatial (18 3, 27 2) sub-configuration of the Gray configuration whose Menger graph is the Pappus graph It is perhaps of interest to note another relationship between the Gray graph and the Pappus graph Namely, the edges of the subdivision of the Pappus graph in Figure 5 can be interpreted as lines of the Gray configuration and the 18 vertices of the Pappus graph are distinguished points of the Gray configuration By deleting 9 points of the Gray configuration we obtain a geometric (18 3, 27 2 ) configuration that produces an unusual drawing of the Pappus graph in 3-space The Pappus graph is a skeleton of a body that is obtained from a 2 by 2 cube with two centrally

7 YET ANOTHER LOOK AT THE GRAY GRAPH 91 symmetric cubes removed The vertex labeling of the Pappus graph in the left side of Figure 8 defines a representation [26] depicted in the right side of the figure In this note we focused on the Gray graph B(3) The natural question to ask is:is there a polytopal generalization that works for some other Bouwer graphs B(n, k)? The Gray graph has surprisingly many appearances in constructions It can be verified, for instance, that it may be obtained from the generalized quadrangle W(3) as defined in [13] p84 This is a semisymmetric tetravalent graph on 80 vertices of girth 8 In other words, W(3) is bipartite, regular, edge-transitive but not vertex-transitive graph There are 54 vertices at distance 4 from each edge and the graph induced on these 54 vertices is the Gray graph Acknowledgements The author would like to acknowledge fruitful discussions with Marston Conder and Branko Grünbaum that led to significant improvement of this note A question posed by Jürgen Bokowski about the graph W(3) led to the discovery that W(3) contains 160 induced copies of the Gray graph Remarks made by the referees were also quite useful and helped to clarify several ideas References [1] N L Biggs and A T White, Permutation Groups and Combinatorial Structures, LMS Lecture Note Series 33, Cambridge University Press, 1979 [2] M Boben and T Pisanski, Polycyclic configurations, Europ J Combinatorics, 24 (2003) [3] M Boben, B Grünbaum, T Pisanski and A Žitnik, Small triangle-free configurations of points and lines, Discrete Comput Geom 35 (2006) [4] J A Bondy and U S R Murty, Graph Theory with Applications New York: North Holland, p 235, 1976 [5] C P Bonnington and T Pisanski, On the orientable genus of the cartesian product of a complete regular tripartite graph with an even cycle, Ars Comb 70 (2004) [6] I Z Bouwer, An edge but not vertex transitive cubic graph, Bull Can Math Soc 11 (1968), [7] I Z Bouwer et al, editors, The Foster Census The Charles Babbage Research Centre, Winnipeg, Canada, 1988 [8] I Z Bouwer, On edge but not vertex transitive regular graphs, J Combin Theory (B) 12 (1972) [9] M Conder, I Hubard and T Pisanski, Constructions for chiral polytopes, Journal London Math Soc, to appear [10] M Conder, A Malnič, D Marušič, T Pisanski and P Potočnik, The edge-transitive but not vertex-transitive cubic graph on 112 vertices, J Graph Theory, 50 (2005) [11] H S M Coxeter, Self-dual configurations and regular graphs, Bull Amer Math Soc 56, (1950) [12] Du, Shaofei, Wang, Furong, Zhang and Lin, An infinite family of semisymmetric graphs constructed from affine geometries, European J Combin 24 (2003), [13] C Godsil and G Royle, Algebraic Graph Theory, Springer 2001 [14] B Grünbaum, Regularity of graphs, complexes and designs, Problèmes combinatoires et théorie des graphes (Colloq Internat CNRS, Univ Orsay, Orsay, 1976), Colloq Internat CNRS, 260, CNRS, Paris, 1978, pp [15] B Grünbaum, Astral (n k ) configrations, Geombinatorics 3 (1993), [16] B Grünbaum, Configurations, Math 553B Special Topics in Geometry, University of Washington, 1999 [17] W Imrich and S Klavžar, Product Graphs Wiley-Interscience Series in Discrete Mathematics and Optimization, 2000

8 92 TOMAŽ PISANSKI1 [18] A A Ivanov, and M E Iofinova, Biprimitive cubic graphs, Investigations in the algebraic theory of combinatorial objects, , Vsesoyuz Nauchno-Issled Inst Sistem Issled, Moscow, (1985) (2002), [19] A Malnič, D Marušič, P Potočnik, and C Wang, An Infinite Family of Cubic Edge- but Not Vertex-Transitive Graphs, Discr Math 280, (2002) [20] D Marušič and T Pisanski, The Gray graph revisited, J Graph Theory, 35 (2000) 1 7 [21] D Marušič and T Pisanski, Symmetries of hexagonal molecular graphs on the torus, Croat Chem Acta, 73 (2000) [22] D Marušič, T Pisanski and S Wilson, The genus of the Gray graph is 7, European J Combin 26, (2005), [23] B Monson, T Pisanski, E Schulte and A Ivić Weiss, Semisymmetric graphs from polytopes, J Combin Theory, Ser A, 114 (2007) [24] T Pisanski (ed), Vega Version 02; Quick Reference Manual and Vega Graph Gallery, IMFM, Ljubljana, [25] T Pisanski and M Randić, Bridges between geometry and graph theory, in Geometry at work, MAA Notes, 53, Math Assoc America, Washington, DC, (2000), [26] T Pisanski and A Žitnik, Representations of graphs and maps, to appear [27] S Wilson, A worthy family of semisymmetric graphs, Discrete Math 271 (2003), Tomaž Pisanski University of Ljubljana and University of Primorska IMFM Jadranska Ljubljana SLOVENIA TomazPisanski@fmfuni-ljsi