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arXiv · 2304.01925

A geometric view on planar graphs and its application to coloring

Henning Wunderlich
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Abstract

While planar graphs are flat from a topological viewpoint, we observe that they are not from a geometric one. We prove that every planar graph can be embedded into a surface consisting of spheres, glued together in a tree-like fashion. As a technical ingredient we prove a statement implying an inverse of the Jordan curve theorem. This statement helps to identify cycles in the planar graph, corresponding to circles of latitude on the spheres. The tree-like embedding then allows for an inductive construction of a four-coloring of the planar graph. Hence, this yields a simple proof of the Four-Color Theorem.

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Henning Wunderlich. 2023-04-02. A geometric view on planar graphs and its application to coloring. https://arxiv.org/abs/2304.01925

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