A simplified proof of the Johansson-Molloy Theorem using the Rosenfeld counting method

TL;DR

This paper presents a simplified proof of the Johansson-Molloy Theorem, demonstrating that triangle-free graphs with maximum degree Δ have chromatic numbers bounded by approximately Δ divided by log Δ.

Contribution

It introduces a simplified proof technique using the Rosenfeld counting method for the Johansson-Molloy Theorem.

Findings

Triangle-free graphs with maximum degree Δ have chromatic number at most (1+o(1))Δ/ log Δ

The proof simplifies previous approaches to the theorem

The result confirms the asymptotic bound on chromatic number for such graphs

Abstract

We show that any triangle-free graph with maximum degree $\Delta$ has chromatic number at most $\left(1+o(1)\right)\Delta/\log \Delta.$