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

TL;DR

This paper extends existing results on the local structure of polyhedral maps on surfaces with non-positive Euler characteristic, showing that large maps must contain vertices with near non-negative curvature, which bounds their path transferability.

Contribution

It generalizes previous work to surfaces with Euler characteristic ≤ 0, establishing a vertex curvature property and a bound on path transferability for large polyhedral maps.

Findings

Maps with more than 126|χ(M)| vertices have vertices with nearly non-negative curvature

Path transferability of such graphs is at most 12

Extends results from the 2-sphere to surfaces with non-positive Euler characteristic

Abstract

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.