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 classified recently by Feng and Zhou (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 $\mathrm{B}\mathrm{C}\mathrm{a}\mathrm{y}(H,S),$ where $S$ is a subset of $H,$ the vertex set of which consists of two copies of $H,$ say $H_0$ and $H_1,$ and the edge set is $\{\{h_0,g_1\}:h,g\in H,g{h}^{-1}\in S\}$. A bi-Cayley graph $\mathrm{B}\mathrm{C}\mathrm{a}\mathrm{y}(H,S)$ is called a BCI-graph if for any bi-Cayley graph $\mathrm{B}\mathrm{C}\mathrm{a}\mathrm{y}(H,T),$ $\mathrm{B}\mathrm{C}\mathrm{a}\mathrm{y}(H,S)\cong \mathrm{B}\mathrm{C}\mathrm{a}\mathrm{y}(H,T)$ implies that $T=h{S}^{\alpha }$ for some $h\in H$ and $\alpha \in $\mathrm{A}\mathrm{u}\mathrm{t}$(H)$\mathrm{.}Itisalsoshownthateverycubicconnectedarc-transitive$0$-type bi-Cayley graph over an abelian group is a BCI-graph.