Cubic factor-invariant graphs of bialternating cycle quotient type

TL;DR

This paper classifies a new class of cubic, vertex-transitive graphs called bialternating cycle quotient type, revealing an infinite family with specific girth properties and Cayley graph structure.

Contribution

It completes the classification of factor-invariant cubic graphs of alternating cycle quotient type by introducing and analyzing the bialternating cycle quotient type.

Findings

Identifies a new infinite 5-parameter family of graphs

Shows these graphs have girth at most 10

Proves they are Cayley graphs with respect to three involutions

Abstract

In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph $\Gamma$ and a vertex-transitive subgroup $G$ of $\mathrm{Aut}(\Gamma)$, a $2$-factor $\mathcal{C}$ of $\Gamma$ is said to be {\em $G$-invariant} if the set $\mathcal{C}$ is preserved by each element of $G$. Investigations of factor-invariant cubic graphs therefore contribute to the rapidly growing theory on cubic vertex-transitive graphs, providing a better insight into the structure of such graphs. Initially, the examples where $\mathcal{C}$ consists of a single or just two cycles were analyzed. In a recent paper by Brian Alspach and the author of this paper, the investigation of the examples for which the corresponding quotient graph $\Gamma_\mathcal{C}$ of $\Gamma$ with respect to $\mathcal{C}$ is a cycle was initiated. Moreover, the graphs of the so-called {\em alternating cycle quotient type} were classified. In this paper, the remaining examples, that is the graphs of the {\em bialternating cycle quotient type}, are classified. It is shown that they belong to a previously unknown infinite $5$-parametric family of graphs of girth at most $10$ and that they are Cayley graphs of groups with respect to three involutions.