Finding the chromatic number by means of critical graphs

We propose a new algorithm for finding the chromatic number of a graph G. The algorithm attempts to determine the smallest possible induced subgraph Gʹ, of G which has the same chromatic number as G. Such a subgraph is said critical since all proper induced subgraphs of Gʹ have a chromatic number strictly smaller than Gʹ. The proposed method is particularly helpful when a k-coloring of anon-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 demonstrate that the new proposed algorithm can solve larger problems than previous known exact methods.