Visibility in Hypercubes

TL;DR

This work studies mutual-visibility in hypercubes, establishing that $μ(Q_n)$ is a constant fraction of $2^{n}$, hence $μ(Q_n)=Θ(2^{n})$, with a concrete lower bound $μ(Q_n) > \frac{14}{75}\cdot 2^{n}$. It also shows that the chromatic mutual-visibility number $χ_{μ}(Q_n)$ is unbounded yet grows only as $O(\log\log n)$, answering a question of prior authors in the negative. Additionally, the paper derives tight asymptotic bounds for the total mutual-visibility number $μ_{t}(Q_n)$, including special cases when $n=2^{m}-1$, and discusses generalizations to Hamming graphs and related coloring variants. The results combine layer-based constructions with forbidden-daisy extremal theory, Ramsey-type arguments, and the Lovász Local Lemma to obtain both concrete constructions and probabilistic colorings, yielding insights with potential applications in network visibility and coding theory.

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 $μ(G)$ 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 $2^{n}/\sqrt{n}\leqμ(Q_{n})\leq2^{n-1}$. In this paper, we prove that $μ(Q_{n})>0.186\cdot2^n$ and thus establish that $μ(Q_{n})=Θ(2^{n})$. We also consider the chromatic mutual-visibility number, $χ_μ(G)$, that is the smallest number of colors used on vertices of $G$, such that every color class is a mutual-visibility set. Klavžar, Kuziak, Valenzuela-Tripodoro, and Yero asked whether it is true that $χ_μ(Q_{n})=O(1)$. We answer their question in the negative by showing that $ω(1)=χ_μ(Q_{n})=O(\log\log{n})$. Finally, we study the so-called total mutual-visibility number of graphs and give asymptotically tight bounds on this parameter for hypercubes.

Figures (7)

• Figure 1: Construction of a dense mutual-visibility set
• Figure 2: A shortest $A,B$-path going through "holes" in the layers
• Figure 3: Each color class is a mutual-visibility set
• Figure 4: A shortest $A,B$-path when $|A|=|B|$ and $k=1$
• Figure 5: A shortest $A,B$-path when $|A|=|B|$ and $k\geq2$

Theorems & Definitions (20)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Lemma 2.1
• proof : Proof of Lemma \ref{['lem:lem2']}
• proof : Proof of Theorem \ref{['thm2']}