Summary: The Chromatic Number
of Random Regular Graphs
and Cristopher Moore2
Microsoft Research, Redmond, WA 98052, USA
University of New Mexico, NM 87131, USA
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 d-regular graph is k, k + 1, or k + 2.
In , 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 .
Significantly less is known for random d-regular graphs Gn,d. In , Frieze
and Luczak extended the results of  for (G(n, p)) to random d-regular graphs,