We introduce a natural k-coloring algorithm and analyze its performance on random graphs with constant expected degree c (G n;p=c=n). For k = 3 our results imply that almost all graphs with n vertices and 1:923 n edges are 3-colorable. This improves the lower bound on the threshold for random 3-colorability signi cantly and settles the last case of a long-standing open question of Bollobas [5]. We also provide a tight asymptotic analysis of the algorithm. We show that for all k 3, if c k ln k 3=2k then the algorithm almost surely succeeds, while for any > 0, and k suf-ciently large, if c (1 + )k ln k then the algorithm almost surely fails. The analysis is based on the use of dierential equations to approximate the mean path of certain Markov chains. 1
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