A note on the mutual-visibility coloring of hypercubes

TL;DR

This paper investigates the mutual-visibility coloring of hypercubes, establishing that the minimum number of colors needed grows slowly with the dimension, specifically as O(log log n).

Contribution

It proves that the mutual-visibility chromatic number of hypercubes is not bounded, answering a previous open question negatively.

Findings

The mutual-visibility chromatic number of hypercubes is unbounded.

It grows at most logarithmically double-logarithmically with dimension.

The growth rate is O(log log n).

Abstract

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 was proved by the authors 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 $$\omega(1)=\chi_{\mu}(Q_{n})=O(\log\log{n}).$$