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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.08762 |
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| _version_ | 1866909780400930816 |
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| author | Nguyen, Tung Scott, Alex Seymour, Paul |
| author_facet | Nguyen, Tung Scott, Alex Seymour, Paul |
| contents | Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want the paths to be far apart, say at distance at least $c$? One might hope that we can find either $k+1$ paths pairwise far apart, or $k$ sets of bounded radius that separate $S$ and $T$, where the bound on the radius is some $\ell$ that depends only on $k,c$ (the ``coarse Menger conjecture''). We showed in an earlier paper that this is false for all $k\ge 2$ and $c\ge3$. To do so we gave a sequence of finite graphs, counterexamples for larger and larger values of $\ell$ with $k=2$, $c=3$. Our counterexamples contained subdivisions of uniform binary trees with arbitrarily large depth as subgraphs.
Here we show that for any binary tree $T$, the coarse Menger conjecture is true for all graphs that contain no subdivision of $T$ as a subgraph, that is, it is true for graphs with bounded path-width (and, further, for graphs with bounded coarse path-width). This is perhaps surprising, since it is false for bounded tree-width. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_08762 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic structure. V. The coarse Menger conjecture in bounded path-width Nguyen, Tung Scott, Alex Seymour, Paul Combinatorics 05C12, 05C40 Menger's theorem tells us that if $S,T$ are sets of vertices in a graph $G$, then (for $k\ge0$) either there are $k+1$ vertex-disjoint paths between $S$ and $T$, or there is a set of $k$ vertices separating $S$ and $T$. But what if we want the paths to be far apart, say at distance at least $c$? One might hope that we can find either $k+1$ paths pairwise far apart, or $k$ sets of bounded radius that separate $S$ and $T$, where the bound on the radius is some $\ell$ that depends only on $k,c$ (the ``coarse Menger conjecture''). We showed in an earlier paper that this is false for all $k\ge 2$ and $c\ge3$. To do so we gave a sequence of finite graphs, counterexamples for larger and larger values of $\ell$ with $k=2$, $c=3$. Our counterexamples contained subdivisions of uniform binary trees with arbitrarily large depth as subgraphs. Here we show that for any binary tree $T$, the coarse Menger conjecture is true for all graphs that contain no subdivision of $T$ as a subgraph, that is, it is true for graphs with bounded path-width (and, further, for graphs with bounded coarse path-width). This is perhaps surprising, since it is false for bounded tree-width. |
| title | Asymptotic structure. V. The coarse Menger conjecture in bounded path-width |
| topic | Combinatorics 05C12, 05C40 |
| url | https://arxiv.org/abs/2509.08762 |