A counterexample to the coarse Menger conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912188508143616 |
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| author | Nguyen, Tung Scott, Alex Seymour, Paul |
| author_facet | Nguyen, Tung Scott, Alex Seymour, Paul |
| contents | Menger's well-known theorem from 1927 characterizes when it is possible to find $k$ vertex-disjoint paths between two sets of vertices in a graph $G$. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when the $k$ paths are required to be pairwise at some distance at least $d$. The result is known for $k\le 2$, but we will show that it is false for all $k\ge 3$, even if $G$ is constrained to have maximum degree at most three. We also give a simpler proof of the result when $k=2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_06685 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A counterexample to the coarse Menger conjecture Nguyen, Tung Scott, Alex Seymour, Paul Combinatorics 05C12, 05C38 Menger's well-known theorem from 1927 characterizes when it is possible to find $k$ vertex-disjoint paths between two sets of vertices in a graph $G$. Recently, Georgakopoulos and Papasoglu and, independently, Albrechtsen, Huynh, Jacobs, Knappe and Wollan conjectured a coarse analogue of Menger's theorem, when the $k$ paths are required to be pairwise at some distance at least $d$. The result is known for $k\le 2$, but we will show that it is false for all $k\ge 3$, even if $G$ is constrained to have maximum degree at most three. We also give a simpler proof of the result when $k=2$. |
| title | A counterexample to the coarse Menger conjecture |
| topic | Combinatorics 05C12, 05C38 |
| url | https://arxiv.org/abs/2401.06685 |