A local property of polyhedral maps on compact two-dimensional manifolds
The authors show that, for any \(k\), each polyhedral map \(G\) on a compact connected surface with Euler characteristic \(\chi\) either contains a path on \(k\) vertices such that each vertex on the path has degree at most \(k\lfloor (5+\sqrt{49-24\chi})/2\rfloor\), or it contains no path on \(k\) vertices at all. Remarkably, no analogous result holds for any connected subgraphs other than paths.
scientific article; zbMATH DE number 1420218