Polynomial iterative algorithms for coloring and analyzing random graphs

TL;DR

This paper investigates the graph coloring problem on random graphs, identifying phase transitions and proposing a polynomial-time algorithm for coloring in the challenging but solvable region.

Contribution

It introduces a precise analysis of the coloring threshold and clustering phase, and proposes a new polynomial-time algorithm for coloring in the hard but colorable region.

Findings

Graphs with low connectivity are almost always colorable.

A critical connectivity value $c_q$ determines colorability thresholds.

A new polynomial-time algorithm effectively colors graphs within the hard but solvable region.

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. Furthermore, we extended our considerations to the case of single instances showing consistent results. This lead us to propose a new algorithm able to color in polynomial time random graphs in the hard but colorable region, i.e when $c\in [c_d,c_q]$.