Far-apart Erdős--Pósa property of long cycles

2026-07-13

AI Summary

This study investigates the long-cycle Erdős–Pósa property with respect to distance in graphs: for given integers \(k\), \(\ell\), and \(d\), every graph either contains \(k\) cycles, each of length at least \(\ell\) and pairwise at distance greater than \(d\), or admits a vertex set \(X\) of size \(f(k,\ell)\) whose \(g(d)\)-neighborhood intersects all such long cycles. Through structural analysis, ball-covering arguments, and recursive constructions, the work extends the classical Erdős–Pósa result from triangles to arbitrary long cycles and establishes tight bounds: \(f(k,\ell) = O(\ell k \log k)\) and \(g(d) = O(d)\). The linear dependence of \(g\) on \(d\) is optimal, and for fixed \(\ell\), the bound on \(f\) is asymptotically optimal in \(k\), significantly improving upon the previous \(O(k^{18} \operatorname{polylog} k)\) bound for the case \(\ell = 3\).

Abstract

We prove that there exist functions $f:\mathbb N^2\to\mathbb N$ and $g:\mathbb N\to\mathbb N$ such that for all positive integers $k$, $d$, and $\ell\ge3$, every graph $G$ either contains $k$ cycles of length at least $\ell$ that are pairwise at distance greater than $d$, or admits a subset of vertices $X$ with $|X|\le f(k,\ell)$ such that $G-B_G(X,g(d))$ contains no cycle of length at least $\ell$, where $B_G(X,r)$ denotes the ball of radius $r$ around $X$. This generalizes a theorem of Dujmović, Joret, Micek, and Morin (2024), which established the $\ell=3$ case. Moreover, we prove that the theorem holds with $f(k,\ell)\in\mathcal{O}(\ell k\log k)$ and $g(d)\in\mathcal{O}(d)$. The linear bound on $g$ is best possible, while the bound on $f$ is optimal as a function of $k$ for every fixed $\ell$. In particular, for $\ell=3$ our result improves the previous bound of $\mathcal{O}(k^{18}\mathsf{polylog} k)$ by Dujmović et al.

Problem

• Erdős–Pósa property
• long cycles
• graph theory
• vertex deletion
• distance constraints

Innovation

• Erdős–Pósa property
• long cycles
• graph theory
• vertex deletion
• distance constraints