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A counterexample to the coarse Menger conjecture
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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 $k=2$.
Forward citations
Cited by 3 Pith papers
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Asymptotic structure. III. Excluding a fat tree
Any graph lacking a c-fat tree minor can be quasi-isometrically approximated by a graph with line-width bounded in terms of the tree and c.
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Asymptotic structure. V. The coarse Menger conjecture in bounded path-width
The coarse Menger conjecture is true for all graphs of bounded path-width.
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Asymptotic structure. VI. Distant paths across a disc
For planar graphs with all terminals on the outer face, the coarse Menger conjecture holds: either k+1 pairwise far paths exist, or k small connected subgraphs of bounded total diameter block all paths.
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