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, Di Fonso, Di Stefano, Navarra, and Piselli showed that \(2^{n}/\sqrt{n}\le \mu (Q_{n})\le 2^{n-1}\) . In this paper, we prove that \(\mu (Q_{n})>0.186\cdot 2^n\) and thus establish that \(\mu (Q_{n})=\Theta (2^{n})\) . We also consider the chromatic mutual-visibility number, \(\chi _{\mu }(G)\) , defined as the smallest number of colors used on vertices of G, such that every color class is a mutual-visibility set in G. Klavžar, Kuziak, Valenzuela-Tripodoro, and Yero asked whether \(\chi _{\mu }(Q_{n})=O(1)\) . We answer their question in the negative, namely, we show that \(\chi _{\mu }(Q_{n})\) is a growing function of n. Moreover, we show that \(\chi _{\mu }(Q_{n})=O(\log \log {n})\) . Finally, we study the so-called total mutual-visibility number of graphs and give asymptotically tight bounds on this parameter for hypercubes.
