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arXiv · 2401.06685

A counterexample to the coarse Menger conjecture

Tung Nguyen·Alex Scott·Paul Seymour
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Abstract

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 $k=2$.

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Tung Nguyen, Alex Scott, Paul Seymour. 2024-01-12. A counterexample to the coarse Menger conjecture. https://arxiv.org/abs/2401.06685

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