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Open access

Sándor Szabó and Bogdán Zaválnij

Abstract

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.

Open access

Sándor Szabó and Bogdán Zaválnij

Abstract

The known fact that coloring of the nodes of a graph improves the performance of practical clique search algorithm is the main motivation of this paper. We will describe a number of ways in which an edge coloring scheme proposed in [8] can be used in clique search. The edge coloring provides an upper bound for the clique number. This estimate has a limitation. It will be shown that the gap between the clique number and the upper bound can be arbitrarily large. Finally, it will be shown that to determine the optimal number of colors in an edge coloring is NP-hard.