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

TL;DR

This paper demonstrates polynomial-time algorithms for weighted list-colorings in split graphs, cographs, and bounded tree-width graphs with fixed colors, and establishes hardness results when assumptions are relaxed.

Contribution

It introduces polynomial algorithms for weighted list-colorings in specific graph classes and proves tightness via hardness results for relaxed conditions.

Findings

Polynomial-time algorithms for list-colorings in split graphs, cographs, and bounded tree-width graphs.

Hardness results showing computational difficulty when assumptions are not met.

Extension to edge-coloring and special graph subclasses.

Abstract

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.