Cubic factor-invariant graphs of bialternating cycle quotient type
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917206275653632 |
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| author | Šparl, Primož |
| author_facet | Šparl, Primož |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11067 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cubic factor-invariant graphs of bialternating cycle quotient type Šparl, Primož Combinatorics 05C25, 05C70 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. |
| title | Cubic factor-invariant graphs of bialternating cycle quotient type |
| topic | Combinatorics 05C25, 05C70 |
| url | https://arxiv.org/abs/2601.11067 |