The first gap for total curvatures of planar graphs with nonnegative curvature

TL;DR

This paper establishes a lower bound of 1/12 for the total curvature of planar graphs with nonnegative curvature and classifies the structures that achieve this bound, advancing understanding of geometric properties of such graphs.

Contribution

It proves a minimal total curvature bound for planar graphs with nonnegative curvature and characterizes the structures that attain this bound.

Findings

Total curvature of such graphs is at least 1/12.

Classification of ambient polygonal surfaces achieving the bound.

Provides geometric insights into planar graphs with nonnegative curvature.

Abstract

We prove that the total curvature of a planar graph with nonnegative combinatorial curvature is at least $\frac{1}{12}$ if it is positive. Moreover, we classify the metric structures of ambient polygonal surfaces for planar graphs attaining this bound.