On cubic vertex-transitive non-Cayley graphs of 2-power orders

Abstract

A Cayley graph (or bi-Cayley graph) Γ is a graph which admits a semiregular group G of automorphisms with exactly one orbit (or two orbits, respectively). Cayley graphs form an important class of vertex-transitive graphs. However, not all vertex-transitive graphs are Cayley. A typical example is the Petersen graph. In this paper, we first prove that all cubic vertex-transitive graphs of order 2-powers are bi-Cayley graphs. We then give a sufficient condition for a cubic vertex-transitive graph of order a 2-power to be non-Cayley. As an application, we construct an infinite family of cubic vertex-transitive non-Cayley graphs of order a 2-power.