Arc-transitive cubic abelian bi-Cayley graphs and BCI-graphs

A finite simple graph is called a bi-Cayley graph over a group H if it has a semiregular automorphism group, isomorphic to H, which has two orbits on the vertex set. Cubic vertex-transitive bi-Cayley graphs over abelian groups have been characterized recently by Zhou and Feng (Europ. J. Combin. 36 (2014), 679-693). In this paper we consider the latter class of graphs and select those in the class which are also arc-transitive. Furthermore, such a graph is called 0-type when it is bipartite, and the bipartition classes are equal to the two orbits of the respective semiregular automorphism group. A 0-type graph can be represented as the graph BiCay(H; S), where S is a subset of H; the vertex set of which consists of two copies of H; say H0 and H1; and the edge set is {{h0,11}: h,g  H,gh-1  S}. A bi-Cayley graph BiCay(H,S) is called a BCI-graph if for any bi-Cayley graph BiCay(H,T), BiCay(H,S)  BiCay(H,T) implies that T = hSa for some h  H and α  Aut(H). It is also shown that every cubic connected arc-transitive 0-type bi-Cayley graph over an abelian group is a BCI-graph.