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

Florent Krząkała, Andrea Pagnani, and Martin Weigt
Phys. Rev. E 70, 046705 – Published 29 October 2004

Abstract

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 cq for the transition from the colorable to the uncolorable phase with q colors. We also study the asymptotic thresholds for q1 finding cq=2qlnqlnq1+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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  • Received 30 March 2004

DOI:https://doi.org/10.1103/PhysRevE.70.046705

©2004 American Physical Society

Authors & Affiliations

Florent Krząkała

  • Dipartimento di Fisica, INFM and SMC, Università di Roma “La Sapienza,” P. A. Moro 2, I-00185 Roma, Italy

Andrea Pagnani

  • Institute for Scientific Interchange (ISI), Viale Settimio Severo 65, I-10133 Torino, Italy

Martin Weigt

  • Institute for Theoretical Physics, University of Göttingen, Tammannstrasse 1, D-37077 Göttingen, Germany Institute for Scientific Interchange (ISI), Viale Settimio Severo 65, I-10133 Torino, Italy

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Vol. 70, Iss. 4 — October 2004

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