Optimize up-color domination over k-colorings

Determine the value of the minimum up-color domination number over all k-colorings of a graph, gamma'_k(G)=min{gamma_{uc}(G,c): c is a k-coloring of G}, and identify the colorings attaining this minimum; analogously, compute the minimum up-color domination weight over optimal colorings, omega'(G)=min{omega_{uc}(G,c): c is an optimal coloring of G}, and identify the colorings attaining it.

Background

The paper defines gamma_{uc}(G,c) and omega_{uc}(G,c) for a graph together with a specified coloring. It also introduces graph-level parameters obtained by optimizing over colorings, but leaves the corresponding optimization and characterization questions unresolved. The proposed problem asks for both the numerical optima and the colorings that achieve them.

References

The following problems are proposed as avenues for future research:

Given a graph $G$ and a natural number $k$, determine the value $$\gamma_k'(G)=\min {\gamma_{uc}(G,c)\, /\, c \mbox{ is a $k$--coloring of } G }$$ and indentify the coloring(s) $c'$ such that $\gamma_{uc}(G,c')=\gamma'_k(G)$.

Analogously, compute the weight $$\omega'(G)=\min {\omega_{uc}(G,c)\, /\, c \mbox{ is an optimal coloring of } G }$$ and find the coloring(s) $c'$ such that $\omega_{uc}(G,c')=\omega'(G)$.

— 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, first Problem environment