A local property of polyhedral maps on compact two-dimensional manifolds

Abstract

We prove that each polyhedral map G on a compact two-dimensional manifold M , other than the sphere S 0 , with Euler characteristic χ( M) contains a k-path, a path on k vertices, such that each vertex of this path has, in G, degree at most k⌊(5+ 49−24χ( M) )/2⌋ or does not contain any k-path. For k even this bound is best possible. Moreover, for any connected graph other than a path no similar estimation exists. We also show that for every integer g lying between the genus of K n and the genus of K n+1 there is an embedding of K n into the orientable surface of genus g which has representativity at least 2. An analogous result is true for nonorientable surfaces as well.