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Cubic factor-invariant graphs of bialternating cycle quotient type

Primož Šparl

Abstract

In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph $Γ$ and a vertex-transitive subgroup $G$ of $\mathrm{Aut}(Γ)$, a $2$-factor $\mathcal{C}$ of $Γ$ 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 $Γ_\mathcal{C}$ of $Γ$ 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.

Cubic factor-invariant graphs of bialternating cycle quotient type

Abstract

In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph and a vertex-transitive subgroup of , a -factor of is said to be {\em -invariant} if the set is preserved by each element of . 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 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 of with respect to 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 -parametric family of graphs of girth at most and that they are Cayley graphs of groups with respect to three involutions.
Paper Structure (13 sections, 9 theorems, 54 equations, 5 figures)

This paper contains 13 sections, 9 theorems, 54 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The graphs $\mathcal{X}_b(m,n,a,b,\ell)$
  4. The automorphisms $\rho$, $\alpha$ and $\gamma$
  5. The four types for the $n = 8$ case
  6. The links
  7. The construction
  8. Existence of a vertex-transitive subgroup preserving the partition
  9. Existence of the automorphism $\gamma$
  10. Existence of the automorphism $\alpha$
  11. Sufficiency for vertex transitivity
  12. The examples with $n = 8$
  13. The $(a,b) = (4,1)$ case

Key Result

Lemma 2.1

Let $\Gamma$ be a cubic vertex-transitive graph admitting a vertex-transitive group $G \leq \mathrm{Aut}(\Gamma)$ and a $G$-invariant partition of the edge set of $\Gamma$ into a $2$-factor with at least two components and a $1$-factor. Then $\varphi({v}^*) = {(\varphi(v))}^*$ holds for each $v \in

Figures (5)

  • Figure 1: The Möbius ladder and the prism resulting from the case $m = 7$ and $n = 4$.
  • Figure 2: The four possibilities for the pair $(a,b)$ in the case of $n = 8$.
  • Figure 3: Two presentations of the graph $\mathcal{X}_b(5,12,1,8,7)$.
  • Figure 4: The two cases when $n = 8$ and $(a,b) \in \{(4,1), (1,5)\}$.
  • Figure :

Theorems & Definitions (13)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Proposition 3.4
  • Lemma 4.1
  • Lemma 4.2
  • Proposition 4.3
  • proof
  • Lemma 5.1
  • ...and 3 more