On cubic bi-Cayley graphs of p-groups

A graph is called a Cayley graph (or bi-Cayley graph, respectively) of a group G if it has a group G of automorphisms acting semiregularly on the vertices with exactly one orbit (or two orbits, respectively). It is known every Cayley graph is vertex-transitive. In this paper, we first present a classification of connected cubic non-Cayley vertex-transitive bi-Cayley graphs of a finite p-group H, where p > 3 is a prime and the derived subgroup of H is either cyclic or isomorphic to Zp × Zp. This is then used to give a classification of connected cubic non-Cayley vertex-transitive graphs of order 2p4 for each prime p.