Finite 1-regular Cayley graphs of valency 5 (Q1953660)

Summary: Let \(\Gamma = \operatorname{Cay}(G, S)\) and \(G \leq X \leq \operatorname{Aut}\Gamma\). We say \(\Gamma\) is \((X,1)\)-regular Cayley graph if \(X\) acts regularly on its arcs. \(\Gamma\) is said to be core-free if \(G\) is core-free in some \(X \leq \operatorname{Aut}(\operatorname{Cay}(G, S))\). In this paper, we prove that if an \((X, 1)\)-regular Cayley graph of valency 5 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 1-regular Cayley graphs of valency 5.