Vertex-transitive Graphs of Valency 3

Abstract

A regular graph of valency 1 is necessarily a disjoint union of paths of length 2 and one of valency 2 must be a disjoint union of cycles. However, regular graphs of valency 3 are so many and varied that it seems to be impossible to describe them all. Among regular graphs are the vertex-transitive ones, those on which a group of automorphisms acts transitively on the vertices. Every regular graph of valency 1 is vertex transitive and those of valency 2 are vertex transitive if and only if they are a disjoint union of cycles of the same length. This paper is intended to be a starting point for a possible description of all vertex transitive graphs of valency 3. It describes them in terms of graphs which can be built up from graphs of smaller valency, bipartite graphs and graphs on which simple groups act as groups of automorphisms. In that simple groups have now been classified, this suggests a program for classifying the graphs. My interest in these graphs was stimulated by the conjecture of L. Lovasz [1, page 249] that every connected vertex-transitive graph has a Hamiltonian path. A likely place to look for counterexamples to this conjecture is among graphs of low valency, perhaps among those of valency 3. I am indebted to C. C. Chen for conversations on this problem. In a group acting on a graph of valency 3, the stabilizer of any vertex acts as a group of permutations on the vertices adjacent to the one it fixes and as such either has three orbits of length one, one of length one and one of length two, or one of length three. These three possibilities are dealt with separately in sections 2, 3 and 4. The first is the easiest and the third is part of a more general result in another of my papers [3]; the middle one requires the most space here. The outcome of this paper can be summarized in the following statement: