Total curvature of planar graphs with nonnegative combinatorial curvature

TL;DR

This paper proves that the total curvature of any planar graph with nonnegative combinatorial curvature is quantized in multiples of 1/12, resolving a previously posed mathematical question.

Contribution

It establishes a fundamental quantization property of total curvature in planar graphs with nonnegative combinatorial curvature, answering an open question.

Findings

Total curvature is an integral multiple of 1/12.

The result applies to all planar graphs with nonnegative combinatorial curvature.

It confirms a conjecture posed by T. Reti.

Abstract

We prove that the total curvature of any planar graph with nonnegative combinatorial curvature is an integral multiple of $\frac{1}{12}.$ As a corollary, this answers a question proposed by T. R\'eti.