A Local Property of Large Polyhedral Maps on Compact 2-Dimensional Manifolds

Abstract

We prove that each polyhedral map G on a compact 2-manifold, which has large enough vertices, contains a k-path, a path on k vertices, such that each vertex of it has, in G, degree at most 6k; this bound being best possible for k even. Moreover, if G has large enough vertices of degree >6k, than it contains a k-path such that each its vertex has degree, in G, at most 5k; this bound is best possible for any k.