 
Summary: Annals of Mathematics, 162 (2005), 13351351
The two possible values of the
chromatic number of a random graph
By Dimitris Achlioptas and Assaf Naor*
Abstract
Given d (0, ) let kd be the smallest integer k such that d < 2k log k.
We prove that the chromatic number of a random graph G(n, d/n) is either kd
or kd + 1 almost surely.
1. Introduction
The classical model of random graphs, in which each possible edge on n
vertices is chosen independently with probability p, is denoted by G(n, p). This
model, introduced by Erdos and R´enyi in 1960, has been studied intensively
in the past four decades. We refer to the books [3], [5], [11] and the references
therein for accounts of many remarkable results on random graphs, as well as
for their connections to various areas of mathematics. In the present paper
we consider random graphs of bounded average degree, i.e., p = d/n for some
fixed d (0, ).
One of the most important invariants of a graph G is its chromatic number
(G), namely the minimum number of colors required to color its vertices so
that no pair of adjacent vertices has the same color. Since the mid1970s, work
