Coarse Menger property of quasi-minor excluded graphs and length spaces
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913110636363776 |
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| author | Liu, Chun-Hung |
| author_facet | Liu, Chun-Hung |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_10068 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Coarse Menger property of quasi-minor excluded graphs and length spaces Liu, Chun-Hung Combinatorics Discrete Mathematics Metric Geometry 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. |
| title | Coarse Menger property of quasi-minor excluded graphs and length spaces |
| topic | Combinatorics Discrete Mathematics Metric Geometry |
| url | https://arxiv.org/abs/2605.10068 |