NEW ZEALAND JOURNAL OF MATHEMATICS Volume 36 (2007), 85–92

YET ANOTHER LOOK AT THE

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 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 ; see for instance [4]. It has been shown that the Gray graph is the unique smallest trivalent semi-; 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 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 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 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 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. The Levi graphs B(n, k) in this case admit the following equivalent descriptions: k B(n, k) = Kn(K1,Kn) = Kk(n)(Kn−1,Kn). 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).

...... 3 1. .. 18 2. . 3 3.. .. 2 6. .. 1 9 ...... 1 ...... 3 ...... 27 ...... 27 ...... 9 ...... 1 ...... (9) ...... K1,3 S(K3 ) Gray S(K3,3,3) K9,1 Figure 3. Link figure for the Gray polytope, a regular 4-polytope, of type {3, 6, 3}.

...... 3 ...... 1 ...... 6 ...... 2...... 2 ...... 6...... 1 ...... 6...... 1 ...... 3 ...... 9 ...... 6 ...... 1 ...... (3) ...... K1,3 S(K3 ) S(K3,3) K6,1 Figure 4. The facet of the Gray polytope is a regular 3-polytope, of (3) type {3, 6} whose skeleton is K3 . Its dual is hexagonal embedding of K3,3 in the torus.

...... 18 ...... 1 ...... 3 ...... 2...... 6 ...... 2...... 1 ...... 9...... 1 ...... 18...... 27 ...... 9 ...... 1 ...... K1,18 S(Pappus) S(K3,3,3) K9,1

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, (xi,yj ,zk), {p, q}, r, 1) with r ∈ {p, q}⊂{xi,yj ,zk} and w + i + j + k 6≡ 0 mod 3. Since the polytope is not self-dual, we have to consider it together with its 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 K3,3,3. 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 (xi,yj,zk) we have u and v where {u, v} := W \{i + j + k mod 3} • Rank 1: between w and {xi,yj} we have (xi,yj ,zk) and (xi,yj ,zm) where k and m are congruent mod 3 to the two numbers in W \{i + j + k mod 3} • Rank 2: between (xi,yj ,zk) and xi we have {xi,yj} and {xi,zk}. • Rank 3: between {xi,yj} and 1 we have {xi} and {yj}. 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 almost- complete 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 K3,3 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. 1 90 TOMAZˇ PISANSKI

Figure 7. The hexagonal embedding of the Pappus graph in the torus. Its dual is the triangulation of the torus by K3,3,3.

Figure 8. The vertices of the Pappus graphs are labeled by triples i, j, k that represent triples (xi,yj ,zk). Note that there are 18 triples with the property i + j + k 6≡ 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 (xi,yj ), (xi,zk) or (yj ,zk) 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 (183, 272) 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 (183, 272) 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 with two centrally YET ANOTHER LOOK AT THE GRAY GRAPH 91 symmetric 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] p.84. 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 Gr¨unbaum that led to significant improvement of this note. A question posed by J¨urgen 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.

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TomaˇzPisanski University of Ljubljana and University of Primorska IMFM Jadranska 19 1111 Ljubljana SLOVENIA [email protected]