AI summaryⓘ
The authors studied a variation of Menger's theorem, which usually relates to connecting points in a graph, but here in a broad geometric setting. They focus on sets of graphs that have the 'weak coarse Menger property,' a condition about finding many well-separated paths or a small cover blocking all such paths. Nguyen, Scott, and Seymour showed that all graphs together do not have this property and asked if special graph families do. The authors proved that graphs excluding some fixed minor (a smaller graph that can't be formed by contracting edges) do have this property with nicely controlled functions, and their result extends to related spaces like certain surfaces and groups, making it a more general and precise outcome.
Menger's theoremcoarse geometryminor-closed familylocally finite graphquasi-isometrygraph minorstring graphCayley graphRiemannian surfaceEuler genus
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.