 
Summary: < <
A Sharp Threshold for kColorability
Dimitris Achlioptas1,
* Ehud Friedgut2
1
Department of Computer Science, University of Toronto, Toronto, Ontario,
Canada M5S3G4; email: optas@cs.toronto.edu
2
Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram,
Jerusalem, Israel; email: ehudf@math.huji.ac.il
Recei¨ed 19 September 1997; accepted 4 February 1998
Z .ABSTRACT: Let k be a fixed integer and f n, p denote the probability that the randomk
Z . Z .graph G n, p is kcolorable. We show that for kG3, there exists d n such that for anyk
)0,
d n y d n qZ . Z .k k
lim f n, s1, and lim f n, s0.k kz / z /n nnĒ nĒ
Z .As a result we conclude that for sufficiently large n the chromatic number of G n, drn
is concentrated in one value for all but a small fraction of d)1. 1999 John Wiley & Sons,
Inc. Random Struct. Alg., 14, 63 70, 1999
Key Words: random graphs; coloring; sharp thresholds
