Basic tetravalent oriented graphs with cyclic normal quotients

Abstract

Let $\mathrm{OG}\left(4\right)$ denote the family of all graph-group pairs $(\Gamma ,G)$ where Γ is finite, 4-valent, connected, and G-oriented (G-half-arc-transitive). A subfamily of $\mathrm{OG}\left(4\right)$ has recently been identified as ‘basic’ in the sense that all graphs in this family are normal covers of at least one basic member. In this paper we provide a description of such basic pairs which have at least one G-normal quotient which is isomorphic to a cycle graph. In doing so, we produce many new infinite families of examples and solve several problems posed in the recent literature on this topic. This result completes a research project aiming to provide a description of all basic pairs in $\mathrm{OG}\left(4\right)$.