Abstract
We consider the problem of coloring Erdös-Rényi and regular random graphs of finite connectivity using 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 for the transition from the colorable to the uncolorable phase with colors. We also study the asymptotic thresholds for finding in perfect agreement with rigorous mathematical bounds, as well as the nature of excited states, and give a global phase diagram of the problem.
1 More- Received 30 March 2004
DOI:https://doi.org/10.1103/PhysRevE.70.046705
©2004 American Physical Society