Half-integral Erd\H{o}s-P\'{o}sa property for non-null $S$-$T$ paths

TL;DR

This paper proves that non-null paths in group-labelled graphs, including odd-length paths in undirected graphs, satisfy a half-integral Erdős-Pósa property, balancing between packing and covering such paths.

Contribution

It establishes the half-integral Erdős-Pósa property for non-null $S$-$T$ paths in finite group-labelled graphs, extending the understanding of path packing and covering.

Findings

Non-null $S$-$T$ paths exhibit the half-integral Erdős-Pósa property.

The property applies to paths of odd length in undirected graphs.

A function $f$ bounds the size of vertex sets intersecting all such paths.

Abstract

For a group $\Gamma$, a $\Gamma$-labelled graph is an undirected graph $G$ where every orientation of an edge is assigned an element of $\Gamma$ so that opposite orientations of the same edge are assigned inverse elements. A path in $G$ is non-null if the product of the labels along the path is not the neutral element of $\Gamma$. We prove that for every finite group $\Gamma$, non-null $S$-$T$ paths in $\Gamma$-labelled graphs exhibit the half-integral Erd\H{o}s-P\'osa property. More precisely, there is a function $f$, depending on $\Gamma$, such that for every $\Gamma$-labelled graph $G$, subsets of vertices $S$ and $T$, and integer $k$, one of the following objects exists: a family $\cal F$ consisting of $k$ non-null $S$-$T$ paths in $G$ such that every vertex of $G$ participates in at most two paths of $\cal F$; or a set $X$ consisting of at most $f(k)$ vertices that meets every non-null $S$-$T$ path in $G$. This in particular proves that in undirected graphs $S$-$T$ paths of odd length have the half-integral Erd\H{o}s-P\'osa property.