#### Date of Original Version

12-2008

#### Type

Article

#### Rights Management

This is the accepted version of the article which has been published in final form at http://dx.doi.org/10.1002/rsa.20286

#### Abstract or Description

We consider the problem of generating a coloring of the random graph _{n,p} uniformly at random using a natural Markov chain algorithm: the Glauber dynamics. We assume that there are *β**Δ* colors available, where *Δ* is the maximum degree of the graph, and we wish to determine the least *β* = *β*(*p*) such that the distribution is close to uniform in *O*(*n* log *n*) steps of the chain. This problem has been previously studied for _{n,p} in cases where *n**p* is relatively small. Here we consider the “dense” cases, where *n**p* ε [*ω* ln *n*, *n*] and *ω* = *ω*(*n*) ∞. Our methods are closely tailored to the random graph setting, but we obtain considerably better bounds on *β*(*p*) than can be achieved using more general techniques.

#### DOI

10.1002/rsa.20286

#### Published In

Random Structures and Algorithms, 36, 3, 251-272.