A Cayley graph Σ=Cay(G,S) is called 1-regular core-free if G is core-free in some Y⩽AutΣ and AutΣ acts regularly on the set of 1-arcs of Σ. In this paper, we classify the 14-valent 1-regular core-free Cayley graphs. In particular, we discover a non-normal Cayley graph on a non-abelian simple group. That is, 14-valent 1-regular Cayley graph on the alternating group A6, with full automorphism group isomorphic to S7. To our knowledge, this is the first example of a non-normal Cayley graph on a non-abelian simple group with even valency greater than 10.
Yang et al. (Sat,) studied this question.