Three-coloring triangle-free graphs on surfaces VII. A linear-time algorithm

TL;DR

This paper presents efficient algorithms for determining 3-colorability of triangle-free graphs on fixed surfaces, including a linear-time decision algorithm and a quadratic-time coloring algorithm, with additional features for vertex coloring constraints.

Contribution

It introduces the first linear-time algorithm for deciding 3-colorability of triangle-free graphs on fixed surfaces and a quadratic-time algorithm for constructing such colorings.

Findings

Linear-time decision algorithm for 3-colorability.

Quadratic-time algorithm for producing a 3-coloring.

Ability to prescribe colors for a bounded set of vertices.

Abstract

We give a linear-time algorithm to decide 3-colorability of a triangle-free graph embedded in a fixed surface, and a quadratic-time algorithm to output a 3-coloring in the affirmative case. The algorithms also allow to prescribe the coloring of a bounded number of vertices.