A Coloring Algorithm for Triangle-Free Graphs

TL;DR

This paper presents a randomized coloring algorithm for triangle-free graphs that uses near-optimal colors and runs efficiently, providing an algorithmic proof of a known upper bound on the chromatic number.

Contribution

It introduces a new randomized algorithm that properly colors triangle-free graphs with near-optimal colors, matching theoretical bounds.

Findings

Uses O(Δ(G)/log Δ(G)) colors, which is optimal up to a constant factor.

Runs in O(nΔ^2(G)log Δ(G)) time with high probability.

Provides an algorithmic proof of the sharp upper bound on the chromatic number of triangle-free graphs.

Abstract

We give a randomized algorithm that properly colors the vertices of a triangle-free graph G on n vertices using O(\Delta(G)/ log \Delta(G)) colors, where \Delta(G) is the maximum degree of G. The algorithm takes O(n\Delta2(G)log\Delta(G)) time and succeeds with high probability, provided \Delta(G) is greater than log^{1+{\epsilon}}n for a positive constant {\epsilon}. The number of colors is best possible up to a constant factor for triangle-free graphs. As a result this gives an algorithmic proof for a sharp upper bound of the chromatic number of a triangle-free graph, the existence of which was previously established by Kim and Johansson respectively.