Minimum coloring problem via semi-tensor product method

Abstract

This paper considers the minimum coloring problem by using the matrix semi-tensor product, and obtains a number of new results and algorithms. Firstly, the minimum coloring problem is expressed into a kind of optimization problem taking in an algebraic form of matrices, based on which an algorithm is designed to find all the minimum coloring schemes for any simple graph. Secondly, an equivalent problem of minimum coloring problem is studied, and a necessary and sufficient condition is proposed, from which a new algorithm to find all the minimum coloring schemes is established. Finally, the effectiveness of the results/algorithms presented in this paper is shown by one illustrative example.