A counterexample to the coarse Menger conjecture

TL;DR

The paper shows that the coarse analogue of Menger's theorem fails for all $k\ge3$, even when graphs have maximum degree at most $3$. It constructs a recursive, binary-tree-based counterexample $G_k$ with carefully arranged $S,T$ such that there are no $k$ $S$-$T$ paths at pairwise distance at least $d$ (specifically $d=3$) and no small separator of size $|X|\le k-1$ that brings all $S$-$T$ paths within a bounded distance $\ell$ of $X$. A simpler combinatorial proof for the $k=2$ case is provided via interval methods, highlighting the contrast between the two regimes. The results motivate weakened conjectures and open questions, including bounded-degree refinements and alternative formulations, and connect to broader themes in coarse graph theory and related works.

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 $k=2$.

Figures (6)

• Figure 1: The dotted curves represent long paths.
• Figure 2: The graph $G_k$. (In this figure, $k=6$.) The vertex $b$ is some vertex on the path between $a_1$ and $s_1$, different from and nonadjacent to $s_1$; it need not be adjacent to $a_1$, and it might equal $a_1$. Its child on that path is $c$.
• Figure 3: Intervals in standard form (the curves join the ends of the intervals).
• Figure 4: For the proof of \ref{['getdisjt']}.
• Figure 5: The output of \ref{['mainint']}. $a_3, b_1$ have been drawn in the same place because we do not know which is larger.