Minimum Coloring k-Colorable Graphs in Polynomial Average Time

We present algorithms for minimum coloring k-colorable graphs drawn from random and semi-random models. In the first model, each allowed edge is included with indpendent probability p. In the second model, an adversary is given the power to vary the edge probability as the random instance is built. Semi-random models were introduced as a way of striking a balance between random graphs and worst-case adversaries. Our algorithms run in polynomial time on the average. Minimum coloring is harder than k-coloring because even a ''short certificate'' is not presently known for the optimality of a coloring