Coloring random graphs

Abstract

We present an algorithm for coloring random 3-chromatic graphs with edge probabilities below the n ^−1/2 “barrier”. Our (deterministic) algorithm succeeds with high probability to 3-color a random 3-chromatic graph produced by partitioning the vertex set into three almost equal sets and selecting an edge between two vertices of different sets with probability p≥n ⁻ 3/5+ε. The method is extended to k-chromatic graphs, succeeding with high probability for p≥n ^−α+ε with α=2k/((k−l)(k+2)) and ε>0. The algorithms work also for Blum's balanced semi-random G_SB(n,p,k) model where an adversary chooses the edge probability up to a small additive noise p. In particular, our algorithm does not rely on any uniformity in the degree.

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