Let be a connected Cayley graph of group G, then Γ is called normal if the right regular representation of G is a normal subgroup of , the full automorphism group of Γ. For the case whereG is a finite nonabelian simple group and Γ is symmetric cubic Cayley graph, Caiheng Li and Shangjin Xu proved that Γis normal with only two exceptions. Since then, the normality of nonsymmetric cubic Cayley graph of nonabelian simple group aroused strong interest of people. So far such graphs which have been known are all normal. Then people conjecture that all of such graphs are either normal or the Cayley subset consists of involutions. In this paper we give an negative answer by two counterexamples. As far as we know these are the first examples for the non-normal cubic nonsymmetric Cayley graphs of finite nonabelian simple groups.
S. J. Xu, X. G. Fang, J. Wang and M. Y. Xu, “5-Arc Transitive Cubic Cayley Graphs on Finite Simple Groups,” European Journal of Combinatorics, Vol. 28, No. 3, 2007, pp. 1023-1036. doi:10.1016/j.ejc.2005.07.020