Long cycles have the edge-Erd\H{o}s-P\'osa property

TL;DR

This paper proves that the set of long cycles in a graph exhibits the edge-Erdős-Pósa property, meaning either many edge-disjoint long cycles exist or a small edge set intersects all such cycles, answering a longstanding open question.

Contribution

It establishes the edge-Erdős-Pósa property for long cycles, providing bounds and resolving a question posed by Birmelé, Bondy, and Reed.

Findings

Proves the edge-Erdős-Pósa property for long cycles.

Provides bounds on the size of the edge set intersecting all long cycles.

Answers a previously open question in graph theory.

Abstract

We prove that the set of long cycles has the edge-Erd\H{o}s-P\'osa property: for every fixed integer $\ell\ge 3$ and every $k\in\mathbb{N}$, every graph $G$ either contains $k$ edge-disjoint cycles of length at least $\ell$ (long cycles) or an edge set $X$ of size $O(k^2\log k + \ell k)$ such that $G-X$ does not contain any long cycle. This answers a question of Birmel\'e, Bondy, and Reed (Combinatorica 27 (2007), 135--145).