Basic tetravalent oriented graphs with cyclic normal quotients

Let OG(4) denote the family of all graph-group pairs (Γ,G) where Γ is finite, 4-valent, connected, and G-oriented (G-half-arc-transitive). A subfamily of OG(4) 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 OG(4).