On the Erd\H{o}s-P\'osa property for long holes in $C_4$-free graphs

TL;DR

This paper establishes an Erdős-Pósa type result for long holes in $C_4$-free graphs, showing a quadratic-logarithmic bound on the size of a vertex set that intersects all long holes or the existence of many disjoint long holes.

Contribution

It proves a bound on the Erdős-Pósa property for holes of length at least 6 in $C_4$-free graphs, answering a question posed by Kim and Kwon.

Findings

Existence of a function $f(k)=O(k^2 \log k)$ for the Erdős-Pósa property.

Either $k$ disjoint holes of length at least 6 or a small vertex set hitting all such holes.

Addresses a previously open question in graph theory.

Abstract

We prove that there exists a function $f(k)=\mathcal{O}(k^2 \log k)$ such that for every $C_4$-free graph $G$ and every $k \in \mathbb{N}$, $G$ either contains $k$ vertex-disjoint holes of length at least $6$, or a set $X$ of at most $f(k)$ vertices such that $G-X$ has no hole of length at least $6$. This answers a question of Kim and Kwon [Erd\H{o}s-P\'osa property of chordless cycles and its applications. JCTB 2020].