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

Published 15 Sep 2017 in math.CO | (1709.05309v1)

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.

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