Random Graph Coloring - a Statistical Physics Approach

TL;DR

This paper applies statistical physics methods to analyze the vertex coloring problem in random graphs, providing exact solutions for 2-coloring and approximate solutions for multiple colors, comparing these with simulations.

Contribution

It introduces an exact analytical solution for 2-coloring and replica symmetric approximations for p-coloring in random graphs, enhancing understanding of graph coloring thermodynamics.

Findings

Exact solution for 2-coloring problem

Replica symmetric approximations for p-coloring

Comparison with enumeration and Monte-Carlo results

Abstract

The problem of vertex coloring in random graphs is studied using methods of statistical physics and probability. Our analytical results are compared to those obtained by exact enumeration and Monte-Carlo simulations. We critically discuss the merits and shortcomings of the various methods, and interpret the results obtained. We present an exact analytical expression for the 2-coloring problem as well as general replica symmetric approximated solutions for the thermodynamics of the graph coloring problem with p colors and K-body edges.