Summary: The concentration of the chromatic number of random graphs
We prove that for every constant > 0 the chromatic number of the random graph G(n, p)
with p = n-1/2-
is asymptotically almost surely concentrated in two consecutive values. This
implies that for any < 1/2 and any integer valued function r(n) O(n
) there exists a function
p(n) such that the chromatic number of G(n, p(n)) is precisely r(n) asymptotically almost surely.
Let G(n, p) denote the random graph on n labeled vertices in which every edge is chosen randomly
and independently with probability p = p(n). We say that G(n, p) has a property A asymptotically
almost surely (a.a.s.) if the probability it satisfies A tends to 1 as n tends to infinity.
One of the most interesting early discoveries in the study of random graphs is that of the fact
that many natural graph invariants are highly concentrated. One of the first striking results of this
type was proved by Matula  and strengthened by various researchers; for fixed values of p almost
all graphs G(n, p) have the same clique number. The proof of this result is not difficult, and is based
on the second moment method.
In this paper we study the concentration of the chromatic number of the random graph G(n, p).