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

Florent Krzakala·Andrea Pagnani·Martin Weigt
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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.

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Florent Krzakala, Andrea Pagnani, Martin Weigt. 2004-07-28. Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs. https://doi.org/10.1103/physreve.70.046705

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