The first gap for total curvatures of 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.

Much of this article was previously contained as arXiv:1703.04119v1, which was split into 1703.04119v2 (and any subsequent version) and this present article