Visibility in Hypercubes

Abstract: A subset $M$ of vertices in a graph $G$ is a mutual-visibility set if any two vertices $u$ and $v$ in $M$ ``see'' each other in $G$, that is, there exists a shortest $u,v$-path in $G$ that contains no elements of $M$ as internal vertices. The mutual-visibility number $\mu(G)$ of a graph $G$ is the largest size of a mutual-visibility set in $G$. Let $n\in\mathbb{N}$ and $Q_{n}$ be an $n$-dimensional hypercube. Cicerone, Fonso, Di Stefano, Navarra, and Piselli showed that $2^{n}/\sqrt{n}\leq\mu(Q_{n})\leq2^{n-1}$. In this paper, we prove that $\mu(Q_{n})>0.186\cdot2^n$ and thus establish that $\mu(Q_{n})=\Theta(2^{n})$. We also consider the chromatic mutual-visibility number, $\chi_{\mu}(G)$, that is the smallest number of colors used on vertices of $G$, such that every color class is a mutual-visibility set. Klav\v{z}ar, Kuziak, Valenzuela-Tripodoro, and Yero asked whether it is true that $\chi_{\mu}(Q_{n})=O(1)$. We answer their question in the negative by showing that $\omega(1)=\chi_{\mu}(Q_{n})=O(\log\log{n})$. Finally, we study the so-called total mutual-visibility number of graphs and give asymptotically tight bounds on this parameter for hypercubes.