Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs

@article{Krzakala2004ThresholdVS,
  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={Physical review. E, Statistical, nonlinear, and soft matter physics},
  year={2004},
  volume={70 4 Pt 2},
  pages={046705}
}
We consider the problem of coloring Erdös-Rényi 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 transition from the colorable to the uncolorable phase with q colors. We also… CONTINUE READING
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