American Physical Society, Physical review E: Statistical, nonlinear, and soft matter physics, 4(70)
DOI: 10.1103/physreve.70.046705
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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. ; Comment: 23 pages, 10 figures. Replaced with accepted version