Coloring precolored perfect graphs

We consider the question of the computational complexity of coloring perfect graphs with some precolored vertices. It is well known that a perfect graph can be colored optimally in polynomial time. Our results give a sharp border between the polynomial and NP-complete instances, when precolored vertices occur. The key result on the polynomially solvable cases includes a good characterization theorem on the existence of an optimal coloring of a perfect graph where a given stable set is precolored with only one color. The key negative result states that the 3-colorability of a graph whose odd circuits go through two fixed vertices is NP-complete. The polynomial algorithms use Grötschel, Lovász and Schrijver's algorithm for finding a maximum clique in a graph, but are otherwise purely combinatorial. c © 1997 John Wiley & Sons, Inc. J Graph Theory 25: