On 1-Arc-regular Graphs

A graph is 1-arc-regular if its full automorphism group acts regularly on the set of its arcs. In this paper, we investigate 1-arc-regular graphs of prime valency, especially of valency 3. First, we prove that if G is a soluble group then a (G, 1)-arc-regular graph must be a Cayley graph of a subgroup of G. Next we consider trivalent Cayley graphs of a finite nonabelian simple group and obtain a sufficient condition under which one can guarantee that Cay(G, S) is 1-arc-regular. Finally, as an application of the result, we construct two infinite families of 1-arc-regular trivalent Cayley graphs with insoluble automorphism groups and, in particular, one of the families is not a Cayley graph.