Finding the chromatic number by means of critical graphs

Abstract

We propose a new exact algorithm for finding the chromatic number of a graph G . The algorithm attempts to determine the smallest possible induced subgraph G' of G which has the same chromatic number as G . Such a subgraph is said critical since all proper induced sub-graph of G' have a chromatic number strictly smaller than G' .The proposed method is particularly helpful when a k -coloring of a non-critical graph is known, and it has to be proved that no ( k -1)-coloring of G exists. Computational experiments on random graphs and on DIMACS benchmark problems demonstate that the new proposed algorithm can solve larger problem than previous known exact methods.