Pentavalent 2-regular core-free Cayley graphs

Abstract

A Cayley graph $\Gamma =\mathrm{Cay}(G,S)$ is said to be 2-regular core-free if G is core-free in some $X⩽\mathrm{Aut}\Gamma$ and $\mathrm{Aut}\Gamma$ acts regularly on the set of 2-arcs of Γ. In this paper, we classify the pentavalent 2-regular core-free Cayley graphs. As a byproduct, we provide another proof of one of the results by Du et al. (2017) [6] regarding pentavalent symmetric graphs over non-abelian simple groups. Namely, we prove that the pentavalent 2-regular Cayley graphs over non-abelian simple groups are normal. Furthermore, we construct a pentavalent core-free 2-transitive Cayley graph $\mathrm{Cay}(G,S)$ such that $\mathrm{Aut}(G,S)$ is transitive but not 2-transitive on S. This answers a question posed by Li in 2008.