Local structures in polyhedral maps on surfaces, and path transferability of graphs

We extend Jendrol' and Skupie\'n'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 $\chi (\M) \le 0$ has more than $126|\chi (\M)|$ 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.