Between coloring and list-coloring: μ-coloring

A new variation of the coloring problem, mu-coloring, is defined in this paper. A coloring of a graph G = (V, E) is a function f : V -> N such that f(v) not equal f(w) if v is adjacent to w. Given a graph G = (V, E) and a function mu : V -> N, G is mu-colorable if it admits a coloring f with f(v) <= mu(v) for each v is an element of V. It is proved that mu-coloring lies between coloring and list-coloring, in the sense of generalization of problems and computational complexity. Furthermore, the notion of perfection is extended to mu-coloring, giving rise to a new characterization of cographs. Finally, a polynomial time algorithm to solve mu-coloring for cographs is shown.