A counterexample to the coarse Menger conjecture

TL;DR

This paper disproves a conjecture extending Menger's theorem to coarse distances for path disjointness in graphs, showing it fails for all cases where k is three or more, even with degree constraints.

Contribution

It provides the first counterexample to the coarse Menger conjecture for all k ≥ 3 and offers a simplified proof for the case k=2.

Findings

Counterexample for k ≥ 3 cases

Fails even with maximum degree 3

Simpler proof for k=2

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 $k=2$.