Finite 1-Regular Cayley Graphs of Valency 5

Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency . 1. Introduction We assume that all graphs in this paper are finite, simple, and undirected. Let be a graph. Denote the vertex set, arc set, and full automorphism group of by , , and , respectively. A graph is called -vertex-transitive or -arc-transitive if acts transitively on or , where . is simply called vertex-transitive, arc-transitive for the case where . In particular, is called -regular if acts regularly on its arcs and then 1-regular when . Let be a finite group with identity element . For a subset of with , the Cayley graph of (with respect to ) is defined as the graph with vertex set such that are adjacent if and only if . It is easy to see that a Cayley graph has valency , and it is connected if and only if . Li proved in [1] that there are only finite number of core-free -transitive Cayley graphs of valency for and and that, with the exceptions and , every -transitive Cayley graph is a normal cover of a core-free one. It was proved in [2] that there are core-free -transitive cubic Cayley graphs up to isomorphism, and there are no core-free -regular cubic Cayley graphs. A natural problem arises. Characterize -transitive Cayley graphs, in particular, which graphs are -regular? Until now, the result about -regular graphs mainly focused constructing examples. For example, Frucht gave the first example of cubic -regular graph in [3]. After then, Conder and Praeger constructed two infinite families of cubic -regular graphs in [4]. Maru？i？ [5] and Malni？ et al. [6] constructed two infinite families of tetravalent -regular graphs. Classifying such graphs has aroused great interest. Motivated by above results and problem, we consider -regular Cayley graphs of valency 5 in this paper. A graph can be viewed as a Cayley graph of a group if and only if contains a subgroup that is isomorphic to and acts regularly on the vertex set. For convenience, we denote this regular subgroup still by . If contains a normal subgroup that is regular and isomorphic to , then is called an X-normal Cayley graph of ; if is not normal in but has a subgroup which is normal in and semiregular on with exactly two orbits, then is called an X-bi-normal Cayley graph; furthermore if , is called normal or bi-normal. Some