Tight bound for the Erdős-Pósa property of tree minors
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910841558794240 |
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| author | Dujmović, Vida Joret, Gwenaël Micek, Piotr Morin, Pat |
| author_facet | Dujmović, Vida Joret, Gwenaël Micek, Piotr Morin, Pat |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06370 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tight bound for the Erdős-Pósa property of tree minors Dujmović, Vida Joret, Gwenaël Micek, Piotr Morin, Pat Combinatorics Discrete Mathematics 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. |
| title | Tight bound for the Erdős-Pósa property of tree minors |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2403.06370 |