Randomly coloring sparse random graphs with fewer colors than the maximum degree

• Published in: Random structures & algorithms Vol. 29; no. 4; pp. 450 - 465
• Main Authors: Dyer, Martin, Flaxman, Abraham D., Frieze, Alan M., Vigoda, Eric
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.12.2006
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20129

We analyze Markov chains for generating a random k‐coloring of a random graph Gn,d/n. When the average degree d is constant, a random graph has maximum degree Θ(log n/log log n), with high probability. We show that, with high probability, an efficient procedure can generate an almost uniformly random k‐coloring when k = Θ(log log n/log log log n), i.e., with many fewer colors than the maximum degree. Previous results hold for a more general class of graphs, but always require more colors than the maximum degree. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006