Sharpness of Cartesian-product palette-index bounds

Determine whether the upper bounds \(\check{s}(C_s \Box G) \le \check{s}(G)+2\) and \(\check{s}(P_s \Box G) \le \check{s}(G)+2\), established for every nontrivial regular graph \(G\) and integer \(s\ge 3\), are sharp.

Background

The paper studies the palette index $\check{s}(G)$, defined as the minimum number of distinct vertex palettes over all proper edge colorings of a graph. In its final section, it derives upper bounds for Cartesian products in which one factor is a cycle or a path and the other factor is a nontrivial regular graph.

Specifically, the paper proves that $\check{s}(C_s \Box G)\leq \check{s}(G)+2$ and $\check{s}(P_s \Box G)\leq \check{s}(G)+2$ for $s\geq 3$. The authors explicitly leave unresolved whether either bound can be attained in general, or whether the bounds can always be improved.