A new variation of the coloring problem, μ-coloring, is defined in this paper. A coloring of a graph G = (V,E) is a function f : V → N such that f(v) 6= f(w) if v is adjacent to w. Given a graph G = (V,E) and a function μ : V → N, G is μ-colorable if it admits a coloring f with f(v) ≤ μ(v) for each v ∈ V . It is proved that μ-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 μ-coloring, giving rise to a new characterization of cographs. Finally, a polynomial time algorithm to solve μ-coloring for cographs is shown.