A construction of vertex-transitive non-Cayley graphs

We present a new construction of infinite families of (finite as well as infinite) vertex-transitive graphs that are not Cayley graphs; many of these turn out even to be arc-transitive. The construction based on representing vertex-transitive graphs as coset graphs of groups, and on a simple but powerful necessary arithmetic condition for Cayley graphs. Vertex-transitive graphs are interesting from both combinatorial as well as grouptheoretical point of view, and have been studied extensively for more than a. century. Although the well-known Cayley graphs have played a prominent role here, there has been an increasing interest in the other side of the fence that is, in vertextransitive graphs that are not Cayley graphs (we borrow the acronym VTNCG for such objects from [8]). By [6] , the problem of constructing VTNCG's is equivalent to the widely studied problem of the existence of certain permutation groups that do not have regular subgroups. And yet, only a few infinite families of VTNCG's have been described in the literature before 1980. The situation has changed dramatically since then; a great deal of the activity was prompted by MarusiC's question [3] of characterizing the values of n for which there exists a VTNCG on n vertices. This resulted in quite a variety of constructions of infinite families of VTNCG's; the most recent ones appear in [4] together with a Australasian Journal of Combinatorics !Q( 1994 L pp. 105-114 large bibliography of other constructions, The general question of characterizing all VTNCG's is probably beyond our reach in the foreseeable future, but much progress has been done for orders that have only a few prime factors (see again [4] for New constructions of infinite families of VTNCG's are therefore of growing interest. Basically, there seem to be two main approaches to the problem. The first assumes that we have enough information on the automorphism group of a given graph to show that it is transitive and cannot contain a regular subgroup. Examples with this property are mostly found among graphs that are related to some of the wellknown families of finite groups, and most of the constructions listed or cited in [4] would fall in this category. The second approach consists in trying to reveal (without invoking the automorphism group) some structural conditions that a Cayley graph has to satisfy, and then show that these are not met by a particular class of vertextransitive graphs. For such necessary conditons and corresponding constructions the reader is referred to [1, 2, 8]. In this paper we present a new construction of infinite families of (Fnite as well as infinite) VTNCG's. Moreover, imposing additional conditions we even obtain arc-transitive non-Cayley graphs (ATNCG's, for short). In a way, our method is a combination of the above ones. First, we represent a vertex-transitive graph by means of a suitable coset graph as in [5, 7, 9] (this is, in fact, equivalent to having a certain information about the structure of some transitive subgroup of the automorphism group of the graph). Then, we show that under some restrictions, our coset graphs do not satisfy a simple but efficient necessary condition for Cayley graphs, given in [1].