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For a graph $G$ with a given list assignment $L$ on the vertices, we give analgebraical description of the set of all weights $w$ such that $G$ is$(L,w)$-colorable, called permissible weights. Moreover, for a graph $G$ with agiven 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 assignmentproblem}. Furthermore, we describe the set of solutions to the {\sl on callproblem}: when $w$ is not a permissible weight, we find all the nearestpermissible weights $w'$. Finally, we give a solution to the non-recoloringproblem keeping a given subcoloring.

Vectorial solutions to list multicoloring problems on graphs