A Branch-and-Cut Algorithm for Equitable Coloring based on a Formulation by Representatives

Abstract

Let G = (V,E) be an undirected graph, where V is the set of vertices and E is that of edges. An equitable k-coloring of G is a partition of V into k disjoint stable subsets such that the difference on the cardinalities of any two subsets is at most one. Each subset is associated with a color and called a color set. The Equitable Coloring Problem (ECP) consists of finding the minimum value of k such that there is an equitable k-coloring of G. This number is said to be the equitable chromatic number of G and it is denoted by χ=(G). The equitable coloring problem was first introduced in [7], motivated by an application to municipal garbage collection [9]. It was proved to be NPhard in [5]. A branch-and-cut algorithm based on an integer programming