Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs
Disordered Systems and Neural Networks
2007-05-23 v2 Statistical Mechanics
Computational Complexity
Abstract
We consider the problem of coloring Erdos-Renyi and regular random graphs of finite connectivity using q colors. It has been studied so far using the cavity approach within the so-called one-step replica symmetry breaking (1RSB) ansatz. We derive a general criterion for the validity of this ansatz and, applying it to the ground state, we provide evidence that the 1RSB solution gives exact threshold values c_q for the q-COL/UNCOL phase transition. We also study the asymptotic thresholds for q >> 1 finding c_q = 2qlog(q)-log(q)-1+o(1) in perfect agreement with rigorous mathematical bounds, as well as the nature of excited states, and give a global phase diagram of the problem.
Cite
@article{arxiv.cond-mat/0403725,
title = {Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs},
author = {Florent Krzakala and Andrea Pagnani and Martin Weigt},
journal= {arXiv preprint arXiv:cond-mat/0403725},
year = {2007}
}
Comments
23 pages, 10 figures. Replaced with accepted version