On tetravalent half-arc-transitive graphs of girth 5

Abstract

A subgroup of the automorphism group of a graph Γ is said to be half-arc-transitive on Γ if its action on Γ is transitive on the vertex set of Γ and on the edge set of Γ but not on the arc set of Γ. Tetravalent graphs of girths 3 and 4 admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth 5. We show that, with two exceptions, all such graphs only have directed 5-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed 5-cycles, study some of their graph theoretic properties and prove that the 5-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth 5 admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth 5.