Bi-primitive 2-arc-transitive bi-Cayley graphs

Abstract

A bipartite graph \(\Gamma \) is a bi-Cayley graph over a group H if \(H\leqslant \textrm{Aut}\Gamma \) acts regularly on each part of \(\Gamma \). A bi-Cayley graph \(\Gamma \) is said to be a normal bi-Cayley graph over H if \(H\unlhd \textrm{Aut}\Gamma \), and bi-primitive if the bipartition preserving subgroup of \(\textrm{Aut}\Gamma \) acts primitively on each part of \(\Gamma \). In this paper, a classification is given for 2-arc-transitive bi-Cayley graphs which are bi-primitive and non-normal.