Finite edge-transitive oriented graphs of valency four with cyclic normal quotients

TL;DR

This paper classifies finite four-valent edge-transitive graphs with cyclic normal quotients, revealing unbounded cyclic quotients and identifying five infinite families with specific restrictions, advancing understanding of such symmetric graphs.

Contribution

It introduces a new classification of four-valent edge-transitive graphs with cyclic normal quotients, including the discovery of five infinite families and analysis of their properties.

Findings

Number of cyclic G-normal quotients can be unbounded

Existence of independent cyclic G-normal quotients imposes restrictions

Identified five infinite families of such graphs

Abstract

We study finite four-valent graphs Gamma admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on Gamma, and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show on the one hand that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph Gamma and we classify all examples. We show there are five infinite families of such pairs (Gamma, G), and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.