The edge-Erd\H{o}s-P\'osa property

TL;DR

This paper investigates the edge-Erdős-Pósa property, revealing that it does not hold for certain non-planar graphs, including large subcubic trees and ladders, thus answering a previously open question.

Contribution

It demonstrates the failure of the edge-Erdős-Pósa property for specific classes of graphs, expanding understanding of graph minor and packing properties.

Findings

Edge-Erdős-Pósa property fails for large subcubic trees.

Edge-Erdős-Pósa property fails for long ladders.

Answers a question by Raymond, Sau, and Thilikos.

Abstract

Robertson and Seymour proved that the family of all graphs containing a fixed graph $H$ as a minor has the Erd\H{o}s-P\'osa property if and only if $H$ is planar. We show that this is no longer true for the edge version of the Erd\H{o}s-P\'osa property, and indeed even fails when $H$ is an arbitrary subcubic tree of large pathwidth or a long ladder. This answers a question of Raymond, Sau and Thilikos.