Between coloring and list-coloring: μ-coloring

Abstract

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) is different from 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 in V. It is proved that mu-coloring lies between coloring and list-coloring, in the sense of generalization of problems and computational complexity. Besides, 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.