Coarse Menger property of quasi-minor excluded graphs and length spaces

2026-05-11

AI Summary

This work investigates whether locally finite graph classes excluding a fixed minor satisfy the weak coarse Menger property: for any vertex subsets, either there exist $k$ mutually distant paths connecting them, or all such connecting paths can be covered by few balls of bounded radius. By integrating graph minor theory, coarse geometric techniques, and analysis of quasi-isometry invariants, the paper establishes that these graph classes indeed possess this property. Specifically, the covering function $f$ depends only on $k$, while the radius function $g$ grows linearly in the scale parameter $r$ and is independent of $k$, achieving optimal asymptotic order. The result extends to length spaces including Riemannian surfaces, chordal graphs, and Cayley graphs, resolving an open problem posed by Nguyen et al.

Abstract

Menger's theorem is an important building block of numerous results in the study of graph structure. We consider a variant in terms of coarse geometry. We say that a set of graphs has the weak coarse Menger property if there exist functions $f$ and $g$ such that for any graph $G$ in this set, subsets $X$ and $Y$ of vertices of $G$, and positive integers $k$ and $r$, either there exist $k$ paths between $X$ and $Y$ pairwise at distance at least $r$, or there exists a union of at most $f(k,r)$ balls of radius at most $g(k,r)$ intersecting all paths between $X$ and $Y$. Nguyen, Scott and Seymour proved that the set of all graphs does not have the weak coarse Menger property and asked whether every proper minor-closed family of finite graphs has it. In this paper, we provide a positive answer to this question in a stronger form: it is true for the set of locally finite graphs with an excluded finite minor, and the functions $f$ and $g$ can be chosen so that $f$ only depends on the number $k$ of the paths in the packing and the function $g$ is a linear function of the distance threshold $r$ and is independent of $k$, which is optimal up to a constant factor. Our result extends to every length space quasi-isometric to a locally finite graph or metric graph with an excluded finite minor, such as complete Riemannian surfaces of finite Euler genus, string graphs, and Cayley graphs of finitely generated minor-excluded groups.

Problem

• coarse Menger property
• minor-excluded graphs
• quasi-isometry
• length spaces
• graph minors

Innovation

• coarse Menger property
• minor-excluded graphs
• quasi-isometry
• length spaces
• graph separators