Total mutual-visibility in Hamming graphs
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914188138381312 |
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| author | Bujtás, Csilla Klavžar, Sandi Tian, Jing |
| author_facet | Bujtás, Csilla Klavžar, Sandi Tian, Jing |
| contents | If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $μ_{\rm t}(G)$ of $G$. In this paper the total mutual-visibility number is studied on Hamming graphs, that is, Cartesian products of complete graphs. Different equivalent formulations for the problem are derived. The values $μ_{\rm t}(K_{n_1}\,\square\, K_{n_2}\,\square\, K_{n_3})$ are determined. It is proved that $μ_{\rm t}(K_{n_1} \,\square\, \cdots \,\square\, K_{n_r}) = O(N^{r-2})$, where $N = n_1+\cdots + n_r$, and that $μ_{\rm t}(K_s^{\,\square\,, r}) = Θ(s^{r-2})$ for every $r\ge 3$, where $K_s^{\,\square\,, r}$ denotes the Cartesian product of $r$ copies of $K_s$. The main theorems are also reformulated as Turán-type results on hypergraphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_05168 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Total mutual-visibility in Hamming graphs Bujtás, Csilla Klavžar, Sandi Tian, Jing Combinatorics If $G$ is a graph and $X\subseteq V(G)$, then $X$ is a total mutual-visibility set if every pair of vertices $x$ and $y$ of $G$ admits a shortest $x,y$-path $P$ with $V(P) \cap X \subseteq \{x,y\}$. The cardinality of a largest total mutual-visibility set of $G$ is the total mutual-visibility number $μ_{\rm t}(G)$ of $G$. In this paper the total mutual-visibility number is studied on Hamming graphs, that is, Cartesian products of complete graphs. Different equivalent formulations for the problem are derived. The values $μ_{\rm t}(K_{n_1}\,\square\, K_{n_2}\,\square\, K_{n_3})$ are determined. It is proved that $μ_{\rm t}(K_{n_1} \,\square\, \cdots \,\square\, K_{n_r}) = O(N^{r-2})$, where $N = n_1+\cdots + n_r$, and that $μ_{\rm t}(K_s^{\,\square\,, r}) = Θ(s^{r-2})$ for every $r\ge 3$, where $K_s^{\,\square\,, r}$ denotes the Cartesian product of $r$ copies of $K_s$. The main theorems are also reformulated as Turán-type results on hypergraphs. |
| title | Total mutual-visibility in Hamming graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2307.05168 |