Packing Topological Minors Half-Integrally
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arXiv
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| Format: | Preprint |
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2017
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| _version_ | 1866912154099122176 |
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| author | Liu, Chun-Hung |
| author_facet | Liu, Chun-Hung |
| contents | The packing problem and the covering problem are two of the most general questions in graph theory. The Erdős-Pósa property characterizes the cases when the optimal solutions of these two problems are bounded by functions of each other. Robertson and Seymour proved that when packing and covering $H$-minors for any fixed graph $H$, the planarity of $H$ is equivalent to the Erdős-Pósa property. Thomas conjectured that the planarity is no longer required if the solution of the packing problem is allowed to be half-integral.
In this paper, we prove that this half-integral version of Erdős-Pósa property holds for packing and covering $H$-topological minors, for any fixed graph $H$, which easily implies Thomas' conjecture. In fact, we prove an even stronger statement in which those topological minors are rooted at any choice of prescribed subsets of vertices.
A number of results on $H$-topological minor free or $H$-minor free graphs have conclusions or requirements tied to properties of $H$. Classes of graphs that can half-integrally pack only a bounded number of $H$-topological minors or $H$-minors are more general topological minor-closed or minor-closed families whose minimal obstructions are more complicated than $H$. Our theorem provides a general machinery to extend those results to those more general classes of graphs without losing their tight connections to $H$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1707_07221 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Packing Topological Minors Half-Integrally Liu, Chun-Hung Combinatorics Discrete Mathematics The packing problem and the covering problem are two of the most general questions in graph theory. The Erdős-Pósa property characterizes the cases when the optimal solutions of these two problems are bounded by functions of each other. Robertson and Seymour proved that when packing and covering $H$-minors for any fixed graph $H$, the planarity of $H$ is equivalent to the Erdős-Pósa property. Thomas conjectured that the planarity is no longer required if the solution of the packing problem is allowed to be half-integral. In this paper, we prove that this half-integral version of Erdős-Pósa property holds for packing and covering $H$-topological minors, for any fixed graph $H$, which easily implies Thomas' conjecture. In fact, we prove an even stronger statement in which those topological minors are rooted at any choice of prescribed subsets of vertices. A number of results on $H$-topological minor free or $H$-minor free graphs have conclusions or requirements tied to properties of $H$. Classes of graphs that can half-integrally pack only a bounded number of $H$-topological minors or $H$-minors are more general topological minor-closed or minor-closed families whose minimal obstructions are more complicated than $H$. Our theorem provides a general machinery to extend those results to those more general classes of graphs without losing their tight connections to $H$. |
| title | Packing Topological Minors Half-Integrally |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/1707.07221 |