On symmetric subgraphs of the 7-cube: an overview

It is shown that the graph QQ) C obtained from the 7 -cube QQ) by deletion of a perfect Hamming code C has a spanning self-complementary subgraph which is edge-transitive but not vertextransitive and also extremal among all the cube subgraphs which are square-blocking and codeavoiding. Our work uses combinatorial techniques involving orientations on the Fano plane and the resulting Steiner triple systems. In this paper, we study some combinatorial properties of a cubic bipartite graph, that we will denote by W, which is edge-transitive but not vertex-transitive. Bouwer mentioned at the end of his introduction in [2] that R.M. Foster found such a graph that has 112 vertices and whose girth is 10. We will see that W has these characteristics. Moreover, we will also see that if C denotes the perfect Hamming code in the 7-cube QQ), lhen W is a spanning self-complementary subgraph of the graph Q\-C In contrast, QO -C is seen to be both an edge-transitive and a vertextransitive graph. The fact thal W and its complementin QQ) -C are edge-transitive but not vertex-transitive subgraphs of Q(7) -C, which is both an edgeand vertextransitive subgraph of the 7-cube (itself edgeand vertex-transitive), testifies to the difficulty of the task of studying the symmetrical properties of subgraphs of n-cubes. The graph Iil appearedin [3] as the subgraph induced by a subset E'ofedges ofthe 7-cube Q0. It was shown in [3] that: (1) E' blocks all the squares of QQ). (2) It C denotes the perfect Hamming code in QQ) gee e.C. [6] ), then no edge of E' has as an endvertex a codeword of C. (3) The cardinality lE' I of E' is the smallest known one among the cardinalities of those edge subsets of QQ) that satisfy (1) and (2). Correspondence tot llalo J. Dejter, Dept. of Math., Univ. of Puerto Rico, Box BF, Rio Piedras, PR 00931-3355, USA. 0012-365X/94/$07.00 O 1994-Elsevier Science B.V. All rights reserved ssDl 0012-365X(92)00128-A 55