A classification of cubic edge-transitive bi-2-metacirculants

Abstract

Let p be a prime. A graph is called a bi-p-metacirculant on a metacyclic p-group H if admits a metacyclic p-group H of automorphisms acting semiregularly on its vertices with two orbits. In this paper, we prove that the complete graph of order 4 and BiCay(H,∅,∅,{1,a,b}), where H=〈a〉×〈b〉≅Z2m×Z2m or H=〈a,b|a4=1,b2=a2,[a,b]=a2〉≅Q8, are the only connected cubic edge-transitive bi-2-metacirculants. This, together with the previous work in Qin and Zhou (2019) [6], completes the classification of cubic edge-transitive bi-p-metacirculants.