Improve color bounds for optimal up-color domination weight colorings

Find better bounds on the number of colors used by an $Omega_{uc}$-domination coloring for general graphs, bipartite graphs, and trees.

Background

The paper proves general bounds on the number of colors relevant to up-color domination and discusses examples for bipartite graphs and trees. It explicitly identifies sharper bounds for OmegaucOmega_{uc}-domination colorings across these three graph classes as unresolved.

References

Find better bounds for the number of colors used by an $\Omega_{uc}$--domination coloring:

\begin{itemize} \item General graphs. \item Bipartite graphs. \item Trees. \end{itemize}

— Domination on Vertex-weighted Graphs Induce by a Coloring  (2502.07248 - Garrido-Vizuete et al., 11 Feb 2025) in Section 5, Conclusions and open problems, fifth Problem environment