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YET ANOTHER LOOK at the GRAY GRAPH Tomaz Pisanski1 in Loving Memory of My Parents the Gray Graph Can Be Constructed by Taking Th

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YET ANOTHER LOOK at the GRAY GRAPH Tomaz Pisanski1 in Loving Memory of My Parents the Gray Graph Can Be Constructed by Taking Th NEW ZEALAND JOURNAL OF MATHEMATICS Volume 36 (2007), 85–92 YET ANOTHER LOOK AT THE GRAY GRAPH Tomaˇz Pisanski1 (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, n flag-transitive non self-dual configurations of type (nn), 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 bi- partite graph K3,3 and, for each edge e of K3,3, 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 edge- but not vertex-transitive graph on 54 vertices is isomorphic to the Gray graph. Let H1 ⊂ H2 ⊂ G be graphs. Define G(H1, H2) to be a bipartite graph defined on all occurrences of H1 as an induced subgraph in G and all occurrences of H2 as an induced subgraph of G and joining an occurrence of H1 to an occurrence of H2 if and only if the occurrence of H1 is an induced subgraph of the corresponding occurrence of H2. The graph G(H1, H2) will be called the Grassmannian of G with respect to H1, H2. 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 B1,B2 and B3. The construc- tions were inspired by Bouwer [6, 8]: Construction 1. Let us consider three disjoint sets on 9 elements: X = {x1, x2, x3}, Y = {y1,y2,y3},Z = {z1,z2,z3} and their union V = X ∪ Y ∪ Z. Let E denote the set of 27 pairs {xi,yj }, {xi,zk}, {yj,zk} and let T denote the set of 27 triples {xi,yj ,zk}. Define the bipartite graph B1 with bipartition (E,T ) in which a pair e ∈ E is adjacent to the triple t ∈ T if and only if e ⊂ t. 1Supported in part by the Public Agency of Research and Development of Slovenia, Grants P1- 0294,J1-6062,L1-7230. 1 86 TOMAZˇ PISANSKI Construction 2. Consider the complete tripartite graph K3,3,3. Let B2 be the graph K3,3,3(K2,K3). 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 K3K3K3. Define B3 to be the graph K3K3K3(K1,K3). 3 We may shorten the notation here: K3,3,3 = K3(3) and K3 K3 K3 = K3 . 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 B1,B2,B3 are all isomorphic to the Gray graph. Proof. To see that B2 is isomorphic to B1, label the vertices in each color class by labels from X, Y , and Z, respectively. The 27 edges of K3,3,3 are naturally labeled by labels from E. Furthermore, the 27 cycles of length 3 are naturally labeled by T and the incidences of B1 correspond precisely to the incidences in B2. Consider the vertex set of K3K3K3 to be X × Y × Z. The labeling can be carried on to B3. The vertices of B3 can be equivalently labeled as sets {xi,yj ,zk} rather than triples (xi,yj,zk). Furthermore a triangle in K3K3K3 is determined by two coordinates: (xi,yj , ∗), (xi, ∗,zk), (∗,yj,zk), or, equivalently by an unordered pair {xi,yj}, {xi,zk}, {yj,zk}. This establishes the isomorphism between B1 and B3 where a pair e ∈ E is mapped to a triangle of B3 and a triple t ∈ T is mapped to a vertex of B3. 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 B3 as the Gray graph. An alternative argument that the three graphs Bi,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. (2) The construction of B1 shows that the graph is edge-transitive. The graph Bi consists of two connected components. One is isomorphic to K3K3K3 in which each edge belongs to exactly one triangle, while the other one does not have this property. Hence Bi 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 K3K3K3 is defined on the vertex set T with triples t,s adjacent if and only if |s ∩ t| = 2. 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 (2564). It is composed of four clearly visible (643, 484) subconfigurations. There is a generalization of our argument from the case n = 3 to general n. 2 Instead of triples, consider n-tuples. Let V (n) be a set {xij | i, j ∈ Zn} of n elements, and let T (n) be the set of all n-tuples whose i th entry is xij for some j, n for all i ∈ Zn. Similarly, let E(n) denote the set of 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 n each (n − 1)-tuple obtained n times. Then define a graph B1(n) on 2n vertices n or, equivalently, a triangle-free (nn) 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 n B2(n)= Kn(n)(Kn−1,Kn) and B3(n)= Kn (K1,Kn), then the graphs B1(n),B2(n) and B3(n) are isomorphic. This generalization implies the existence of geometric triangle-free point- line- and flag-transitive non self-dual configurations (vk) 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 2nn vertices of valence n has girth 8 and diameter 2n, and each vertex corresponding to a n-tuple has (n − 1)n an- tipodal 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 1 88 TOMAZˇ PISANSKI n+1 the wreath product Sn ≀ Sn, of order (n!) , 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]. n The construction of the configurations naturally generalizes not only to (nn), but to configurations (pq,nk) 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 (pq,nk), where p = kq and n = qkq−1. The configuration is self-dual and and only if q = k =2.
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