Coloring the nodes of a directed graph

It is an empirical fact that coloring the nodes of a graph can be used to speed up clique search algorithms. In directed graphs transitive subtournaments can play the role of cliques. In order to speed up algorithms to locate large transitive tournaments we propose a scheme for coloring the nodes of a directed graph. The main result of the paper is that in practically interesting situations determining the optimal number of colors in the proposed coloring is an NP-hard problem. A possible conclusion to draw from this result is that for practical transitive tournament search algorithms we have to develop approximate greedy coloring algorithms.