The analysis of a list-coloring algorithm on a random graph

We introduce a natural k-coloring algorithm and analyze its performance on random graphs with constant expected degree c (G_n,p=cn/). 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 significantly and settles the last case of a long-standing open question of Bollobas. We also provide a tight asymptotic analysis of the algorithm. We show that for all k⩾3, if c⩽k In k-3/2k then the algorithm almost surely succeeds, while for any ε>0, and k sufficiently large, if c⩾(1+ε)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