 
Summary: The Analysis of a ListColoring Algorithm on a Random Graph
(extended abstract)
Dimitris Achlioptas*t
(optasQcs.toronto.edu1
Abstract
We introduce a natural kcoloring algorithm and
analyze its performance on random graphs with con
stant expected degree c (Gn,p=c/n).For k = 3 our re
sults imply that almost all graphs with n vertices and
1.923n edges are 3colorable. This improves the lower
bound on the threshold for random 3colorability sig
nificantly and settles the last case of Q longstanding
open question of Bollobds [5]. We also provide a tight
asymptotic analysis of the algorithm. We show that
for all k 2 3, if c 5 klnk  3/2k then the algorithm
almost surely succeeds, while for any E > 0, and k suf
ficiently large, if c 2 (1+E)k In k then the algorithm
almost surely fails. The analysis is based on the use
of differential equations to approximate the mean path
of certain Markov chains.
