Arc-transitive trivalent Cayley graphs

Abstract

A combinatorial graph Γ is symmetric, or arc-transitive, if its automorphism group acts transitively on the arcs of Γ, and s-arc-transitive (resp. s-arc-regular) if its automorphism group acts transitively (resp. regularly) on the set of s-arcs of Γ, which are the walks of length s in Γ in which any three consecutive vertices are distinct. It was shown by Tutte (1947, 1959) that every finite symmetric trivalent graph is s-arc-regular for some s≤5. Djoković and Miller (1980) took this further by showing that there are seven types of arc-transitive group action on finite trivalent graphs, characterised by the stabilisers of a vertex and an edge. The latter classification was refined by Conder and Nedela (2009), in terms of what types of arc-transitive subgroup can occur in the automorphism group of Γ. In this paper we address the question of when a finite trivalent Cayley graph is arc-transitive, by determining when a connected finite arc-transitive trivalent graph is a Cayley graph. We show that in five of the 17 Conder-Nedela classes, there is no Cayley graph, while in two others, every graph is a Cayley graph. In eight of the remaining ten classes, we give necessary conditions on the order of the graph for it to be Cayley; there is no such condition in the other two. Also we show that in each of those last ten classes, there are infinitely many Cayley graphs and infinitely many non-Cayley graphs.