A subset $M$ of vertices in a graph $G$ is a mutual-visibility set if for any two vertices $u,v\in{ M}$ there exists a shortest $u$-$v$ path in $G$ that contains no elements of $M$ as internal vertices. Let $\chi_{ \mu}(G)$ be the least number of colors needed to color the vertices of $G$, so that each color class is a mutual-visibility set. Let $n\in\mathbb{ N}$ and $Q_{ n}$ be an $n$-dimensional hypercube. It has been shown that the maximum size of a mutual-visibility set in $Q_{ n}$ is at least $\Omega(2^{ n})$. Klav\v{ z}ar, Kuziak, Valenzuela-Tripodoro, and Yero further asked whether it is true that $\chi_{ \mu}(Q_{ n})=O(1)$. In this note we answer their question in the negative by showing that $$\omega(1)=\chi_{ \mu}(Q_{ n})=O(\log\log{ n}).$$
Categories: math.CO