Cubic core-free symmetric m-Cayley graphs

Abstract

An m-Cayley graph $$\Gamma $$ over a group G is defined as a graph which admits G as a semi-regular group of automorphisms with m orbits. This generalises the notions of a Cayley graph (where $$m = 1$$ ) and a bi-Cayley graph (where $$m = 2$$ ). The m-Cayley graph $$\Gamma $$ over G is said to be normal if G is normal in the automorphism group $$\mathrm{Aut}(\Gamma )$$ of $$\Gamma $$ , and core-free if the largest normal subgroup of $$\mathrm{Aut}(\Gamma )$$ contained in G is the identity subgroup. In this paper, we investigate properties of symmetric m-Cayley graphs in the special case of valency 3, and use these properties to develop a computational method for classifying connected cubic core-free symmetric m-Cayley graphs. We also prove that there is no 3-arc-transitive normal Cayley graph or bi-Cayley graph (with valency 3 or more), which answers a question posed by Li (Proc Amer Math Soc 133:31–41 2005). Using our classification method, we give a new proof of the fact that there are exactly 15 connected cubic core-free symmetric Cayley graphs, two of which are Cayley graphs over non-abelian simple groups. We also show that there are exactly 109 connected cubic core-free symmetric bi-Cayley graphs, 48 of which are bi-Cayley graphs over non-abelian simple groups, and that there are 1, 6, 81, 462 and 3267 connected cubic core-free 1-arc-regular 3-, 4-, 5-, 6- and 7-Cayley graphs, respectively.