Towards a conjecture of Birmelé-Bondy-Reed on the Erdős-Pósa property of long cycles

Abstract

A conjecture of Birmelé, Bondy and Reed states that for any integer $\ell \ge 3$, 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.