Coloring Random Graphs

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+e. The method is extended to k-chromatic graphs, succeeding with high probability for p≥n−α+e with α=2k/((k−l)(k+2)) and e>0. The algorithms work also for Blum's balanced semi-random GSB(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.