  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.
