A complete algorithm to solve the graph-coloring problem

The Graph k-Colorability Problem (GCP) is a well known NP-hard problem which consist in finding the k minimum number of colors to paint the vertices of a graph in such a way that any two vertices joined by an edge have always different colors. Many years ago, Simulated Annealing (SA) was used for graph coloring task obtaining good results; however SA is not a complete algorithm and it not always gets the optimal solution. In this paper GCP is transformed into the Satisfiability Problem and then it is solved using a algorithm that uses the Threshold Accepting algorithm (a variant of SA) and the Davis & Putnam algorithm. The new algorithm is a complete one and so it gets better quality that the classical simulated annealing algorithm.