The Erd\H{o}s-P\'osa property for prime-length cycles fails (and beyond)

TL;DR

This paper demonstrates that prime-length cycles and certain sets of cycles do not satisfy the Erdős-Pósa property, even in planar and projective planar graphs, under various density conditions.

Contribution

It generalizes the failure of the Erdős-Pósa property to prime-length cycles and sets with specific density properties, answering a question in graph theory.

Findings

Prime-length cycles lack the t-th Erd
51s-Pf3sa property in planar graphs.

Sets with lower density zero of cycle lengths do not have the t-th Erd
51s-Pf3sa property.

Porous sets of cycle lengths do not satisfy the Erd
51s-Pf3sa property in projective planar graphs.

Abstract

We prove that for every $t \in \mathbb{N}$, prime-length cycles do not have the $\frac{1}{t}$-integral Erd\H{o}s-P\'osa property, even when restricted to planar graphs. We in fact prove a more general density result. For every $t \in \mathbb{N}$ and every subset $L \subseteq \mathbb{N}$ with lower density zero, the set of cycles whose length is in $L$ do not have the $\frac{1}{t}$-integral Erd\H{o}s-P\'osa property, even when restricted to planar graphs. We also consider a less restrictive density condition on $L$, called porous, where the complement of $L$ contains arbitrarily long sequences of consecutive integers. We prove that for every porous set $L \subseteq \mathbb{N}$, the set of cycles whose length is in $L$ do not have the Erd\H{o}s-P\'osa property, even when restricted to projective planar graphs. Our results partially answer a question of Gollin, Hendrey, Kwon, Oum, and Yoo [Math. Ann., 393(2):2507-2559, 2025].