A tighter Erd\"os-P\'osa function for long cycles

TL;DR

This paper establishes a tighter bound on the Erdős-Pósa function for long cycles, showing that the minimum vertex set intersecting all such cycles can be bounded by O(l k log k), improving previous quadratic bounds.

Contribution

It introduces a new bound for the Erdős-Pósa function for long cycles, reducing it from quadratic to near-linear in terms of k, with a dependence on l.

Findings

The function f(k,l) = O(l k log k) bounds the size of vertex sets intersecting all long cycles.

The result improves previous bounds of (l k^2) by Birmele9, Bondy, and Reed.

The theorem applies to all graphs and all natural numbers k and l.

Abstract

We prove that there exists a bivariate function f with f(k,l) = O(l k log k) such that for every naturals k and l, every graph G has at least k vertex-disjoint cycles of length at least l or a set of at most f(k,l) vertices that meets all cycles of length at least l. This improves a result by Birmel\'e, Bondy and Reed (Combinatorica, 2007), who proved the same result with f(k,l) = \Theta(l k^2).