  For a fixed number of colors, we show that, in node-weighted split graphs,
cographs, and graphs of bounded tree-width, one can determine in polynomial
time whether a proper list-coloring of the vertices of a graph such that the
total weight of vertices of each color equals a given value in each part of a
fixed partition of the vertices exists. We also show that this result is tight
in some sense, by providing hardness results for the cases where any one of the
assumptions does not hold. The edge-coloring variant is also studied, as well
as special cases of cographs and split graphs.
