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Tight bound for the Erd\H{o}s-P\'osa property of tree minors

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arxiv 2403.06370 v2 pith:E4CMACPY submitted 2024-03-11 math.CO cs.DM

Vida Dujmović , Gwenaël Joret , Piotr Micek , Pat Morin This is my paper · ORCID
classification math.COcs.DM
keywords boundeveryminortreeverticesbestcontainsearlier
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abstract

Let $T$ be a tree on $t$ vertices. We prove that for every positive integer $k$ and every graph $G$, either $G$ contains $k$ pairwise vertex-disjoint subgraphs each having a $T$ minor, or there exists a set $X$ of at most $t(k-1)$ vertices of $G$ such that $G-X$ has no $T$ minor. The bound on the size of $X$ is best possible and improves on an earlier $f(t)k$ bound proved by Fiorini, Joret, and Wood (2013) with some fast growing function $f(t)$. Moreover, our proof is short and simple.

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  1. Erd\H{o}s-P\'osa property of rooted tree minors

    math.CO 2026-07 accept novelty 7.0 of 10

    Rooted tree minors satisfy a linear Erdős-Pósa bound: k vertex-disjoint rooted models or a hitting set of O(k) vertices.

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