How many colors to color a random graph? Cavity, Complexity, Stability and all that

TL;DR

This paper reviews recent statistical physics approaches to graph coloring, identifying a critical connectivity threshold that separates colorable and uncolorable phases, and discusses the accuracy of the replica symmetry breaking method.

Contribution

It provides a detailed phase diagram and confirms the threshold values for random graph coloring using the one-step replica symmetry breaking ansatz.

Findings

Identifies the threshold at c_q=2q log q - log q - 1 + o(1)

Confirms the accuracy of the replica symmetry breaking method for this problem

Draws a comprehensive phase diagram of the coloring problem

Abstract

We review recent progress on the statiscal physics study of the problem of coloring random graphs with q colors. We discuss the existence of a threeshold at connectivity c_q=2q log q-log q-1+o(1) separting two phases which are respectivily COL(orable) and UNCOL(orable) with q colors; We also argue that the so-called one-step replica symmetry breaking ansatz used to derive these results give it exact threshold values, and draw a general phase diagram of the problem.