Three-coloring triangle-free planar graphs in linear time

TL;DR

This paper presents a simple, efficient linear-time algorithm for 3-coloring triangle-free planar graphs, improving upon previous algorithms that required more complex data structures and quadratic time.

Contribution

It introduces a straightforward linear-time algorithm for 3-coloring triangle-free planar graphs, along with a simpler proof of Grotzsch's theorem.

Findings

Linear-time 3-coloring algorithm developed

Avoids complex data structures for simplicity

Provides a simpler proof of Grotzsch's theorem

Abstract

Grotzsch's theorem states that every triangle-free planar graph is 3-colorable. Several relatively simple proofs of this fact were provided by Thomassen and other authors. It is easy to convert these proofs into quadratic-time algorithms to find a 3-coloring, but it is not clear how to find such a coloring in linear time (Kowalik used a nontrivial data structure to construct an O(n log n) algorithm). We design a linear-time algorithm to find a 3-coloring of a given triangle-free planar graph. The algorithm avoids using any complex data structures, which makes it easy to implement. As a by-product we give a yet simpler proof of Grotzsch's theorem.