Legal coloring of graphs

The following computational problem was initiated by Manber and Tompa (22nd FOCS Conference, 1981) : Given a graph G = (V,E) and a real function f : V→R which is a proposed vertex coloring. Decide whether f is a proper vertex coloring of G. The elementary steps are taken to be linear comparisons. Lower bounds on the complexity of this problem are derived using the chromatic polynomial of G. It is shown how geometric parameters of a space partition associated with G influence the complexity of this problem. In particular we show (theorem 6) a lower bound of (m/2)1/2 log m + O(m1/2), where m is the number of edges of the graph in question. Existing methods for analyzing such space partitions are suggested as a powerful tool for establishing lower bounds for a variety of computational problems. Many interesting open problems are presented.