2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/228552We extend Jendrol' and Skupień's results about the local structure of maps on the 2-sphere: In this paper we show that if a polyhedral map $G$ on a surface $\M$ of Euler characteristic $χ(\M) \le 0$ has more than $126|χ(\M)|$ vertices, then $G$ has a vertex with "nearly" non-negative combinatorial curvature. As a corollary of this, we can deduce that path transferability of such graphs are at most 12.1 table, and 3 figuresCombinatoricsGeometric Topology05C10text