For a graph G with a given list assignment L on the vertices, we give an algebraic description of the set of all weights w such that G is -colorable, called permissible weights. Moreover, for a graph G with a given list L and a given permissible weight w, we describe the set of all -colorings of G. By the way, we solve the channel assignment problem. Furthermore, we describe the set of solutions to the on call problem: when w is not a permissible weight, we find all the nearest permissible weights  Finally, we give a solution to the non-recoloring problem keeping a given subcoloring