Erdős-Pósa property of cycles that are far apart

TL;DR

The paper proves a coarse Erdős–Pósa variant for cycles that are far apart: for every graph $G$ and integers $k,d$, either $G$ contains $k$ cycles with pairwise distance $>d$ or there exists a subset $X$ with $|X|\le f(k)$ such that $G-B_G(X,g(d))$ is a forest. The authors develop a packing-versus-hitting framework built around $d$-unicyclic cycles, employing a BFS-unicycle structure and tools from Simonovits's cycle-packing results and Gyárfás–Lehel Helly-type theorems, to obtain the quantitative bound $f(k)=\mathcal{O}(k^{18}\mathrm{polylog}\,k)$ with $g(d)=19d$. The proof relies on a chain of lemmas (notably grow_unicycle, double_unicycle, and all_the_ys) that iteratively reduce the problem to either a large $d$-packing or a small hitting set whose balls cover the relevant structures, yielding a constructive bound. This work advances the coarse Erdős–Pósa program by linking graph minors, coarse geometry, and unicyclic decomposition, and it includes an appendix establishing polynomial bounds for the key Helly-type function.

Abstract

We prove that there exist functions $f,g:\mathbb{N}\to\mathbb{N}$ such that for all nonnegative integers $k$ and $d$, for every graph $G$, either $G$ contains $k$ cycles such that vertices of different cycles have distance greater than $d$ in $G$, or there exists a subset $X$ of vertices of $G$ with $|X|\leq f(k)$ such that $G-B_G(X,g(d))$ is a forest, where $B_G(X,r)$ denotes the set of vertices of $G$ having distance at most $r$ from a vertex of $X$.

Theorems & Definitions (28)

• Theorem 1
• Theorem 2: Simonovits67
• Theorem 3: gyarfas.lehel:helly
• Lemma 4
• proof
• Lemma 5
• Lemma 6
• proof
• Lemma 7
• proof