Chasing the k-colorability threshold

TL;DR

This paper introduces a novel second moment method inspired by physics to improve bounds on the k-colorability threshold in random graphs, narrowing the gap between rigorous results and physics conjectures.

Contribution

Develops a new approach to the second moment method for graph k-coloring, significantly tightening bounds on the colorability threshold.

Findings

New lower bound within 1.39 of the upper bound

Bounds are within 0.39 of physics conjecture

Improves upon previous bounds that were unbounded in k

Abstract

Over the past decade, physicists have developed deep but non-rigorous techniques for studying phase transitions in discrete structures. Recently, their ideas have been harnessed to obtain improved rigorous results on the phase transitions in binary problems such as random $k$-SAT or $k$-NAESAT (e.g., Coja-Oghlan and Panagiotou: STOC 2013). However, these rigorous arguments, typically centered around the second moment method, do not extend easily to problems where there are more than two possible values per variable. The single most intensely studied example of such a problem is random graph $k$-coloring. Here we develop a novel approach to the second moment method in this problem. This new method, inspired by physics conjectures on the geometry of the set of $k$-colorings, allows us to establish a substantially improved lower bound on the $k$-colorability threshold. The new lower bound is within an additive $2\ln 2+o_k(1)\approx 1.39$ of a simple first-moment upper bound and within $2\ln 2-1+o_k(1)\approx 0.39$ of the physics conjecture. By comparison, the best previous lower bound left a gap of about $2+\ln k$, unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004].