Basic Tetravalent Oriented Graphs of Independent-Cycle Type

TL;DR

This paper classifies basic 4-valent graphs with specific symmetry properties, focusing on those with multiple independent cyclic normal quotients, advancing understanding of their structure and symmetry relations.

Contribution

It provides an explicit classification of basic pairs with at least two independent cyclic G-normal quotients in the family of 4-valent oriented graphs.

Findings

Identified all basic pairs with multiple independent cyclic G-normal quotients.

Established the structure of these graphs as normal covers of simpler quotient graphs.

Enhanced the understanding of symmetry and quotient structures in 4-valent graphs.

Abstract

The family $\mathcal{OG}(4)$ consisting of graph-group pairs $(\Gamma, G)$, where $\Gamma$ is a finite, connected, 4-valent graph admitting a $G$-vertex-, and $G$-edge-transitive, but not $G$-arc-transitive action, has recently been examined using a normal quotient methodology. A subfamily of $\mathcal{OG}(4)$ has been identified as `basic', due to the fact that all members of $\mathcal{OG}(4)$ are normal covers of at least one basic pair. We provide an explicit classification of those basic pairs $(\Gamma, G)$ which have at least two independent cyclic $G$-normal quotients (these are $G$-normal quotients which are not extendable to a common cyclic normal quotient).