A counterexample to the coarse Menger conjecture

Tung Nguyen, Alex D. Scott, Paul D. Seymour

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≤2<math><mi is="true">k</mi><mo is="true">≤</mo><mn is="true">2</mn></math>, but we will show that it is false for all k≥3<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">3</mn></math>, even if G is constrained to have maximum degree at most three. We also give a simpler proof of the result when k=2<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mn is="true">2</mn></math>.