On vertex sets inducing tangles
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910707057950720 |
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| author | Albrechtsen, Sandra von Bergen, Hanno Jacobs, Raphael W. Knappe, Paul Wollan, Paul |
| author_facet | Albrechtsen, Sandra von Bergen, Hanno Jacobs, Raphael W. Knappe, Paul Wollan, Paul |
| contents | Diestel, Hundertmark and Lemanczyk asked whether every $k$-tangle in a graph is induced by a set of vertices by majority vote. We reduce their question to graphs whose size is bounded by a function in $k$. Additionally, we show that if for any fixed $k$ this problem has a positive answer, then every $k$-tangle is induced by a vertex set whose size is bounded in $k$. More generally, we prove for all $k$ that every $k$-tangle in a graph $G$ is induced by a weight function $V(G) \to \mathbb{N}$ whose total weight is bounded in $k$. As the key step of our proofs, we show that any given $k$-tangle in a graph $G$ is the lift of a $k$-tangle in some topological minor of $G$ whose size is bounded in $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13656 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On vertex sets inducing tangles Albrechtsen, Sandra von Bergen, Hanno Jacobs, Raphael W. Knappe, Paul Wollan, Paul Combinatorics 05C40, 05C83, 05C69 Diestel, Hundertmark and Lemanczyk asked whether every $k$-tangle in a graph is induced by a set of vertices by majority vote. We reduce their question to graphs whose size is bounded by a function in $k$. Additionally, we show that if for any fixed $k$ this problem has a positive answer, then every $k$-tangle is induced by a vertex set whose size is bounded in $k$. More generally, we prove for all $k$ that every $k$-tangle in a graph $G$ is induced by a weight function $V(G) \to \mathbb{N}$ whose total weight is bounded in $k$. As the key step of our proofs, we show that any given $k$-tangle in a graph $G$ is the lift of a $k$-tangle in some topological minor of $G$ whose size is bounded in $k$. |
| title | On vertex sets inducing tangles |
| topic | Combinatorics 05C40, 05C83, 05C69 |
| url | https://arxiv.org/abs/2411.13656 |