A note on the mutual-visibility coloring of hypercubes

TL;DR

This note investigates the mutual-visibility coloring number $\chi_{\mu}$ for the $n$-dimensional hypercube $Q_n$. It provides a negative answer to whether $\chi_{\mu}(Q_n)$ can be bounded by a constant, showing instead that $\chi_{\mu}(Q_n)=\omega(1)$ and, more sharply, $\chi_{\mu}(Q_n)=O(\log\log n)$. The lower bound uses a layered subcube argument combined with hypergraph Ramsey numbers to force three equally colored layers in a copy of $Q_{2q}$, which destroys mutual-visibility for that color class. The upper bound leverages a layer-reduction lemma and a probabilistic, Lovász Local Lemma-based construction to color middle layers with two colors, achieving $\chi_{\mu}(Q_n)=O(\log\log n)$. Together, these results show that mutual-visibility coloring can be unbounded yet grows extremely slowly with $n$, separating $\chi_{\mu}$ from the trivial lower bound $|V(Q_n)|/\mu(Q_n)$.

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 $χ_μ(G)$ 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 $Ω(2^{n})$. Klavžar, Kuziak, Valenzuela-Tripodoro, and Yero further asked whether it is true that $χ_μ(Q_{n})=O(1)$. In this note we answer their question in the negative by showing that $$ω(1)=χ_μ(Q_{n})=O(\log\log{n}).$$

Theorems & Definitions (10)

• Theorem 1: axenovich2024visibility
• Theorem 2
• proof : Proof of the lower bound
• Claim 3
• proof : Proof of Claim \ref{['pop']}
• Remark 4
• Lemma 5: axenovich2024visibility
• Lemma 6
• proof : Proof of Lemma \ref{['eye']}
• proof : Proof of the upper bound