We characterize the obstructions to the Erdős-Pósa property of A-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and for every subset <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>, the family of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>-labelled A-paths whose lengths are in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> satisfies the half-integral Erdős-Pósa property. Moreover, we give a characterization of such <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Lambda \subseteq \Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Λ</mi> <mo>⊆</mo> <mi mathvariant="normal">Γ</mi> </mrow> </math></EquationSource> </InlineEquation> for which the same family of A-paths satisfies the full Erdős-Pósa property.

Erdős-Pósa property of A-paths in unoriented group-labelled graphs

期刊论文
发表日期：

2026年3月4日

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O-joung Kwon,
Youngho Yoo

摘要

We characterize the obstructions to the Erdős-Pósa property of A-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $\Gamma $ $Γ$ and for every subset $\Lambda $ $Λ$ of $\Gamma $ $Γ$ , the family of $\Gamma $ $Γ$ -labelled A-paths whose lengths are in $\Lambda $ $Λ$ satisfies the half-integral Erdős-Pósa property. Moreover, we give a characterization of such $\Gamma $ $Γ$ and $\Lambda \subseteq \Gamma $ $Λ \subseteq Γ$ for which the same family of A-paths satisfies the full Erdős-Pósa property.