On the classification of quartic half-arc-transitive metacirculants

A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ρ and σ , where ρ is ( m , n ) -semiregular for some integers m ≥ 1 and n ≥ 2 , and where σ normalizes ρ , cyclically permuting the orbits of ρ in such a way that σ m has at least one fixed vertex. In a recent paper Marušič and the author showed that each connected quartic half-arc-transitive metacirculant belongs to one (or possibly more) of four classes of such graphs, reflecting the structure of the quotient graph relative to the semiregular automorphism ρ . One of these classes coincides with the class of the so-called tightly-attached graphs, which have already been completely classified. In this paper a complete classification of the second of these classes, that is the class of quartic half-arc-transitive metacirculants for which the quotient graph relative to the semiregular automorphism ρ is a cycle with a loop at each vertex, is given.