Long $A$-$B$-paths have the edge-Erd\H os-P\'osa property

TL;DR

This paper establishes that for any fixed number of long $A$-$B$-paths, either the graph contains that many edge-disjoint paths or a bounded edge set intersects all such paths, extending the Erdős-Pósa property to long paths.

Contribution

It proves the edge-Erdős-Pósa property for long $A$-$B$-paths, generalizing previous vertex-based results to the edge setting.

Findings

Existence of a function $f(k, ext{ell})$ bounding the size of a hitting set

Either $k$ edge-disjoint long $A$-$B$-paths exist or a small edge set intersects all such paths

Extension of Erdős-Pósa property to long paths in graphs

Abstract

For a fixed integer $\ell$ a path is long if its length is at least $\ell$. We prove that for all integers $k$ and $\ell$ there is a number $f(k,\ell)$ such that for every graph $G$ and vertex sets $A,B$ the graph $G$ either contains $k$ edge-disjoint long $A$-$B$-paths or it contains an edge set $F$ of size $|F|\leq f(k,\ell)$ that meets every long $A$-$B$-path. This is the edge analogue of a theorem of Montejano and Neumann-Lara (1984). We also prove a similar result for long $A$-paths and long $\mathcal{S}$-paths.