Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.04791 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- 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, Di 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, $χ_μ(G)$, 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 $χ_μ(Q_{n})=O(1)$. We answer their question in the negative, namely, we show that $χ_μ(Q_{n})$ is a growing function of $n$. Moreover, we show that $χ_μ(Q_{n})=O(\log\log{n})$. Finally, we study the so-called total mutual-visibility number of graphs and give asymptotically tight bounds on this parameter for hypercubes.