Short proofs of coloring theorems on planar graphs

TL;DR

This paper presents concise proofs of several classic and new 3-coloring theorems for planar graphs using a recent lower bound on edges in k-critical graphs, simplifying existing complex proofs.

Contribution

It introduces short, elegant proofs for known theorems and a new 3-colorability result for graphs derived from triangle-free planar graphs.

Findings

Short proof of Grötzsch Theorem for triangle-free planar graphs

Proof of Grunbaum-Aksenov Theorem for planar graphs with up to three triangles

New result: graphs from triangle-free planar graphs with a degree-4 vertex are 3-colorable

Abstract

A recent lower bound on the number of edges in a k-critical n-vertex graph by Kostochka and Yancey yields a half-page proof of the celebrated Gr\"otzsch Theorem that every planar triangle-free graph is 3-colorable. In this paper we use the same bound to give short proofs of other known theorems on 3-coloring of planar graphs, among whose is the Gr\"unbaum-Aksenov Theorem that every planar with at most three triangles is 3-colorable. We also prove the new result that every graph obtained from a triangle-free planar graph by adding a vertex of degree at most four is 3-colorable.