For a graph G with a given list assignment L on the vertices, we give an
algebraical description of the set of all weights w such that G is
(L,w)-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
(L,w)-colorings of G. By the way, we solve the {\sl channel assignment
problem}. Furthermore, we describe the set of solutions to the {\sl on call
problem}: when w is not a permissible weight, we find all the nearest
permissible weights w′. Finally, we give a solution to the non-recoloring
problem keeping a given subcoloring.Comment: 10 page