In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and a vertex-transitive subgroup G of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textrm{Aut}(\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Aut</mtext> <mo stretchy="false">(</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, a 2-factor <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> is said to be G-invariant if the set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> is preserved by each element of G. Investigations of factor-invariant cubic graphs therefore contribute to the rapidly growing theory on cubic vertex-transitive graphs, providing a better insight into the structure of such graphs. Initially, the examples where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> consists of a single or just two cycles were analyzed. In a recent paper by Alspach and the author of this paper, the investigation of the examples for which the corresponding quotient graph <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Gamma _\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mi mathvariant="script">C</mi> </msub> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> with respect to <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathcal {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">C</mi> </math></EquationSource> </InlineEquation> is a cycle was initiated. Moreover, the graphs of the so-called alternating cycle quotient type were classified. In this paper, the remaining examples, that is the graphs of the 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

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2026年5月7日

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Primož Šparl

摘要

In 2019, investigation of the so-called factor-invariant cubic graphs was initiated by Alspach, Khodadadpour and Kreher. For a cubic graph $\Gamma $ $Γ$ and a vertex-transitive subgroup G of $\textrm{Aut}(\Gamma )$ $Aut (Γ)$ , a 2-factor $\mathcal {C}$ $C$ of $\Gamma $ $Γ$ is said to be G-invariant if the set $\mathcal {C}$ $C$ 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 $\mathcal {C}$ $C$ consists of a single or just two cycles were analyzed. In a recent paper by Alspach and the author of this paper, the investigation of the examples for which the corresponding quotient graph $\Gamma _\mathcal {C}$ $Γ_{C}$ of $\Gamma $ $Γ$ with respect to $\mathcal {C}$ $C$ is a cycle was initiated. Moreover, the graphs of the so-called alternating cycle quotient type were classified. In this paper, the remaining examples, that is the graphs of the 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.