Erd\H{o}s-P\'osa property of $A$-paths in unoriented group-labelled graphs

TL;DR

This paper characterizes when certain group-labelled paths in graphs satisfy the Erdős-Pósa property, providing conditions for both half-integral and full versions based on group and subset properties.

Contribution

It offers a complete characterization of obstructions and conditions for the Erdős-Pósa property of $A$-paths in unoriented group-labelled graphs, including length restrictions.

Findings

Half-integral Erdős-Pósa property holds for all finite abelian groups and subsets.

Characterization of groups and subsets where the full Erdős-Pósa property is satisfied.

Identification of obstructions to the Erdős-Pósa property in this context.

Abstract

We characterize the obstructions to the Erd\H{o}s-P\'osa property of $A$-paths in unoriented group-labelled graphs. As a result, we prove that for every finite abelian group $\Gamma$ and for every subset $\Lambda$ of $\Gamma$, the family of $\Gamma$-labelled $A$-paths whose lengths are in $\Lambda$ satisfies the half-integral Erd\H{o}s-P\'osa property. Moreover, we give a characterization of such $\Gamma$ and $\Lambda\subseteq\Gamma$ for which the same family of $A$-paths satisfies the full Erd\H{o}s-P\'osa property.