Weighted and locally bounded list-colorings in split graphs, cographs, and partial k-trees

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.