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 ρρ and σσ, where ρρ is (m,n)(m,n)-semiregular for some integers m≥1m≥1 and n≥2n≥2, and where σσ normalizes ρρ, cyclically permuting the orbits of ρρ in such a way that σmσm 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 ρρ. 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 ρρ is a cycle with a loop at each vertex, is given.