A normal quotient analysis for some families of oriented four-valent graphs

TL;DR

This paper investigates the normal quotient structures of specific families of finite, connected, four-valent oriented graphs, revealing new insights into their internal configurations and symmetry properties.

Contribution

It introduces a detailed analysis of normal quotients in these graphs, uncovering unexpected cross-overs and classifying basic graphs based on their normal quotients.

Findings

Identified cross-overs between graph families via normal quotients

Determined which graphs are 'basic' with only degenerate proper normal quotients

Showed that only certain orientations are invariant under vertex- and edge-transitive group actions

Abstract

We analyse the normal quotient structure of several infinite families of finite connected edge-transitive, four-valent oriented graphs. These families were singled out by Marusic and others to illustrate various different internal structures for these graphs in terms of their alternating cycles (cycles in which consecutive edges have opposite orientations). Studying the normal quotients gives fresh insights into these oriented graphs: in particular we discovered some unexpected `cross-overs' between these graph families when we formed normal quotients. We determine which of these oriented graphs are `basic', in the sense that their only proper normal quotients are degenerate. Moreover, we show that the three types of edge-orientations studied are the only orientations, of the underlying undirected graphs in these families, which are invariant under a group action which is both vertex-transitive and edge-transitive.