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

Abstract

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 BiCay(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 ∈ 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 = hS α 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.

Edge-transitive bi-p-metacirculants of valency p

“…Remark 1.6. The classification of cubic edge-transitive bi-Cayley graphs on abelian groups has been given in [10,23]. So our result actually completes the classification of all cubic edge-transitive bi-p-metacirculants for each odd prime p.…”