A tight Erd\H{o}s-P\'osa function for long cycles

TL;DR

This paper extends the Erdős-Pósa theorem to long cycles, establishing a tight bound on the vertex set needed to intersect all such cycles or find many disjoint ones, with implications for graph tree-width.

Contribution

It generalizes Erdős-Pósa to long cycles, providing an optimal bound on vertex sets and linking to graph tree-width, improving prior results.

Findings

Either find k disjoint long cycles or a small vertex set intersecting all such cycles.

The vertex set size bound is tight up to constants.

Results imply bounds on the tree-width of graphs without long cycle packings.

Abstract

A classic result of Erd\H{o}s and P\'osa says that any graph contains either $k$ vertex-disjoint cycles or can be made acyclic by deleting at most $O(k \log k)$ vertices. Here we generalize this result by showing that for all numbers $k$ and $l$ and for every graph $G$, either $G$ contains $k$ vertex-disjoint cycles of length at least $l$, or there exists a set $X$ of $\mathcal O(kl+k\log k)$ vertices that meets all cycles of length at least $l$ in $G$. As a corollary, the tree-width of any graph $G$ that does not contain $k$ vertex-disjoint cycles of length at least $l$ is of order $\mathcal O(kl+k\log k)$. These results improve on the work of Birmel\'e, Bondy and Reed '07 and Fiorini and Herinckx '14 and are optimal up to constant factors.