Between coloring and list-coloring: μ-coloring

A new variation of the coloring problem, μ-coloring, is defined in this paper. Given a graph G and a function μ, a μ-coloring is a coloring where each vertex v of G must receive a color at most μ(v). It is proved that μ-coloring lies between coloring and list-coloring, in the sense of generalization of problems and computational complexity. The notion of perfection is extended for μ-coloring, leading us to a new characterization of cographs. Finally, a polynomial time algorithm to solve μ-coloring for cographs is shown.