Polynomial kernels for pre-weighted vertex-coloring

Establish whether the pre-weighted {0,1}-edge and {1,2}-edge Vertex-Coloring problems admit polynomial kernels when parameterized by the vertex cover number.

Background

The paper proves polynomial kernels for the unweighted Vertex-Coloring problems with edge-weight sets {0,1} and {1,2}, parameterized by vertex cover number. It also introduces pre-weighted variants in which some edge weights are fixed in advance and shows that these variants are fixed-parameter tractable under the same parameter.

Polynomial kernels for the two pre-weighted variants remain unresolved because the marking procedure used for the unweighted kernel constructions breaks down in the presence of pre-weighted edges. The authors explicitly identify the development of alternative kernelization techniques as necessary.

References

The most immediate open question---in our view---is whether Vertex-coloring Pre or Vertex-coloring Pre admit polynomial kernels parameterized by the vertex cover number. Our marking procedure breaks down when there are pre-weighted edges, so other ideas are needed.

— Vertex-Coloring Edge-Weighting: Kernelization and Generalization  (2609.27719 - Aute et al., 23 Sep 2026) in Section 7, paragraph titled “Open problems”

A second question to address is whether we can improve on the running time of 2{O(5{k} \log (k+2))} \cdot n{O(1)} for arbitrary pre-edge-weightings parameterized by the vertex cover number (\autoref{thm:prezo-general}), to bring it closer to the other running times in this paper.

— Vertex-Coloring Edge-Weighting: Kernelization and Generalization  (2609.27719 - Aute et al., 23 Sep 2026) in Section 7, paragraph titled “Open problems”