The Gray graph revisited

Certain graph theoretic properties and alternative de nitions of the Gray graph the smallest known cubic edge but not vertex transitive graph are discussed in detail Introduction history The smallest known cubic edge but not vertex transitive graph has vertices and is known as the Gray graph denoted hereafter by G The rst published account on the Gray graph is due to Bouwer who mentioned that this graph had in fact been discovered by Marion C Gray in thus explaining its name Bouwer gives two ways of constructing G First three copies of the complete bipartite graphK are taken and to a particular edge e of K a vertex is inserted in the interior of e in each of the three copies of K and the resulting three vertices are then joined to a new vertex The second construction identi es a particular Hamilton cycle in G and the corresponding chords see Figure Some other ways of constructing G are presented in this note thus shed ding a new light on the structure of this remarkable graph In the computa tions involved a usage of the Vega package was essential Supported in part by Ministrstvo za znanost in tehnologijo Slovenije proj no P Supported in part by Ministrstvo za znanost in tehnologijo Slovenije proj no J and J Figure The Gray Graph with an identi ed Hamilton cycle as in Structural properties and alternative de nitions The Gray graph G is a cubic bipartite and edge but not vertex transitive graph Its automorphism group AutG acts transitively on each of the bipar tition sets Since its girth is it is the Levi graph of two dual triangle free point line and ag transitive non self dual con gurations There is a simple reason for intransitivity of AutG on the vertex set of G Two vertices have the same distance sequence if and only if they belong to the same bipartition set More precisely the two distance sequences are and and the respective vertices are henceforth called black and white It follows that the diameter of G is Note that jAutGj Let S denote an arbitrary Sylow subgroup of AutG It may be seen that S Z oZ and that S acts transitively on the edge set of G as well as on the sets of black and white vertices We note further that there are a total of octagons that is induced cycles of length in G Octagons in G play an essential role in our Con struction below Let us also mention that each vertex of G is contained in octagons and each edge of G is contained in octagons Construction This construction was pointed to us by Randic in a per sonal communication but see also and gives G in the LCF notation as the graph with code thus identy ng a Hamilton cycle which admits a Z symmetry see Figure Figure The Gray graph with an identi ed Hamilton cycle admitting a Z Figure The Gray graph is a Z regular cover of K The broken lines carry identity voltages whereas Black and white vertices of K lift to the vertices with distance sequences and respectively Figure The Gray graph is a Z regular cover of the Pappus graph P The broken lines carry identity voltages whereas Black and white vertices of P lift to the vertices with distance sequences and respectively Construction Following the construction using three copies of K from the previous section it may be deduced that G is a regular cover of K with Z as the group of covering transformations see for notation and terminology More precisely letting and be the two generators of Z then Figure gives G as a Z regular cover of K We note that by averaging the voltages that is by replacing each and by and by selecting instead of the group h i Z its subgroup h i Z as a voltage group the resulting voltage graph K lifts to a fold cover P on vertices which is isomorphic to the Levi graph of the well known Pappus con guration Note that P is a arc transitive bipartite graph and is the underlying graph of the voltage graph depicted in Figure Furthermore the voltage graph in Figure lifts to the Gray graph G Let us also mention that the normalizer in AutG of the corresponding two groups of covering transformations Z and Z is a subgroup of order Construction Another interesting construction identi es G as the anti line graph of a certain Cayley graph of the Sylow subgroup S of AutG Each element of S can be represented as a quadruple i j k l where i j k l Z and the multiplication obeys the following rules Figure The line graph L G is a Cayley graph for the Sylow subgroup S Z o Z Aut G Note that S acts transitively on the edges and the two bipartition sets of G i j k r s t w i r j s k t w i j k r s t w i t j r k s w i j k r s t w i s j t k r w Note that the normal subgroup in S isomorphic to Z  consists of all the elements i j k S i j k Z Let a and b Then the Cayley graph Cay S fa a b b g of S with respect to the set of generators fa a b b g is the line graph L G of the Gray graph see Figure Figure The Gray graph and ve disjoint octagons Construction Finally we want to discuss by far the most interesting rule for constructing the Gray graph which shows that G possesses a surpris ing feature of containing itself within itself In what follows we shall rst discuss some graph theoretic distinction between the two kinds of vertices and then gradually build our way towards a construction of G identifying the Gray graph within the Gray graph feature Now let x be an arbitrary vertex in G and let G x i ir where i ir denote the subgraph of G induced by all the vertices at distance i ir from x Of course since the vertex orbits of AutG coincide with the color classes the graph G x i ir may only depend on the color of x An unordered triple of octagons may be associated with an arbitrary ver tex in G in the following way It transpires that for a black vertex b the graph G b is isomorphic to a union of three octagons C these octagons give Figure The Gray graph as a union of G G and G rise to the above triple Similarly for a white vertex w the graph G w is isomorphic to a union of three disjoint octagons and four isolated vertices C K Again the three octagons give rise to the above mentioned triple Further it may be seen that the black and white triples have at most one octagon in common which happens if and only if the corresponding two vertices are neighbors in G It thus follows that the graph whose vertex set consists of all the white and black triples of octagons with the adjacency meaning that two triples have non empty intersection is isomorphic to the Gray graph Consequently the graph whose vertex set consists of all oc tagons in G with two octagons adjacent if and only if they both belong to one of the above triples is isomorphic to the line graph L G of the Gray graph Figure identi es ve disjoint octagons the upper three octagons de ne a black triple whereas the top octagon together with the two bottom octagons de ne a white triple Based on the above discussion the rule for constructing G may be given via three auxiliary graphs G G b G G b G G b shown in Figure The root black vertex in G is labeled The six vertices and at distance from are glued to the corresponding vertices of G which is depicted in the projective plane The twelve vertices a a a a b b b b c c c c at distance from are identi ed with the corresponding twelve midpoints of the subdivided cube G Three disjoint grey octagons Ea Eb Ec are visible on the projective plane The faces of the subdivided cube give rise to three opposite disjoint octagonal pairs A A B B C C Figure The local situation in the octagon graph isomorphic to L G Figure depicts the local situation in the above mentioned graph of octagons isomorphic to L G where as seen from the graph G in Figure the adjacency corresponds to two octagons being at distance in G For instance the octagon Ea is at distance from Eb and Ec as well as from octagons C and C see also the ve shaded octagons in Figure