Coloring random graphs

TL;DR

This paper investigates the graph coloring problem on random graphs, identifying thresholds for colorability and analyzing the complex structure of solutions that affect algorithm performance.

Contribution

It determines the critical connectivity thresholds for colorability and characterizes the clustering phase that influences algorithmic complexity.

Findings

Graphs with low connectivity are almost always colorable.

High connectivity graphs are uncolorable.

A clustering phase exists where solutions split into many clusters.

Abstract

We study the graph coloring problem over random graphs of finite average connectivity $c$. Given a number $q$ of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on $q$, we find the precise value of the critical average connectivity $c_q$. Moreover, we show that below $c_q$ there exist a clustering phase $c\in [c_d,c_q]$ in which ground states spontaneously divide into an exponential number of clusters and where the proliferation of metastable states is responsible for the onset of complexity in local search algorithms.