Directed cycles have the edge-Erd\H os-Pósa property

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.