On quartic half-arc-transitive metacirculants

Abstract

Following Alspach and Parsons, a {\em 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 $m \geq 1$, $n \geq 2$, and where $σ$ normalizes $ρ$, cyclically permuting the orbits of $ρ$ in such a way that $σ^m$ has at least one fixed vertex. A {\em half-arc-transitive graph} is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.