 
Summary: The Chromatic Number
of Random Regular Graphs
Dimitris Achlioptas1
and Cristopher Moore2
1
Microsoft Research, Redmond, WA 98052, USA
optas@microsoft.com
2
University of New Mexico, NM 87131, USA
moore@cs.unm.edu
Abstract. Given any integer d 3, let k be the smallest integer such
that d < 2k log k. We prove that with high probability the chromatic
number of a random dregular graph is k, k + 1, or k + 2.
1 Introduction
In [10], Luczak proved that for every real d > 0 there exists an integer k = k(d)
such that w.h.p.1
(G(n, d/n)) is either k or k + 1. Recently, these two possible
values were determined by the first author and Naor [4].
Significantly less is known for random dregular graphs Gn,d. In [6], Frieze
and Luczak extended the results of [9] for (G(n, p)) to random dregular graphs,
