Towards a conjecture of Birmel\'e-Bondy-Reed on the Erd\H{o}s-P\'osa property of long cycles

TL;DR

This paper advances the understanding of the Erdős-Pósa property for long cycles by proving a tighter bound on the size of vertex sets needed to intersect all long cycles in graphs lacking two disjoint such cycles.

Contribution

The authors provide a new proof that reduces the upper bound on the vertex set size from previous results, establishing that at most 3ℓ/2 + 7/2 vertices suffice.

Findings

Improved upper bound on vertex set size for long cycles

Confirmed the conjecture for a tighter bound

Enhanced understanding of the Erdős-Pósa property for long cycles

Abstract

A conjecture of Birmel\'e, Bondy and Reed states that for any integer $\ell\geq 3$, every graph $G$ without two vertex-disjoint cycles of length at least $\ell$ contains a set of at most $\ell$ vertices which meets all cycles of length at least $\ell$. They showed the existence of such a set of at most $2\ell+3$ vertices. This was improved by Meierling, Rautenbach and Sasse to $5\ell/3+29/2$. Here we present a proof showing that at most $3\ell/2+7/2$ vertices suffice.