An asymmetric formulation to the harmonious coloring problem

A proper k-coloring of a graph G is a harmonious coloring if each pair of distinct edges have different sets of colors in their endpoints. The harmonious chromatic number, denoted by h(G), is the minimum $k\in\mathbb{N}$ such that G has a harmonious k-coloring. In this paper, we present an exact integer formulation denoted asymmetric representative formulation and give some computational results for it on random graphs with distinct edge densities.