Directed cycles have the edge-Erd\H os-P\'osa property

TL;DR

This paper proves that for any fixed number of directed cycles, there exists a bounded edge set whose removal destroys all directed cycles, establishing an edge-Erdős-Pósa property for directed cycles.

Contribution

It establishes the edge-Erdős-Pósa property for directed cycles, showing a bounded edge set exists to eliminate all directed cycles if fewer than k are found.

Findings

Existence of a bounded edge set for directed cycles

Edge-Erdős-Pósa property proven for directed cycles

Applicable to all digraphs regardless of size

Abstract

In this short note we prove that for every $k\in \mathbb{N}$ there is a $t_k\in\mathbb{N}$ such that for every digraph $G$ there are either $k$ edge-disjoint directed cycles in $G$ or a set $X$ of at most $t_k$ edges such that $G-X$ contains no directed cycle.