| Title: | Total mutual-visibility in Hamming graphs |
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| Authors: | ID Bujtás, Csilla (Author)
ID Klavžar, Sandi (Author)
ID Tian, Jing (Author) |
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| Files: |  PDF - Presentation file, download (512,21 KB)
MD5: 76C8DA0C8283EF377B0D2CDA5E2781EF
 URL - Source URL, visit https://www.opuscula.agh.edu.pl/om-vol45iss1art5
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | 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 $\mu_{\rm t}(G)$ 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 $\mu_{\rm t}(K_{n_1}\,\square\, K_{n_2}\,\square\, K_{n_3})$ are determined. It is proved that $\mu_{\rm t}(K_{n_1} \,\square\, \cdots \,\square\, K_{n_r}) = O(N^{r-2})$▫, where $N = n_1+\cdots + n_r$, and that $\mu_{\rm t}(K_s^{\,\square\,, r}) = \Theta(s^{r-2})$ 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. |
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| Keywords: | mutual-visibility set, total mutual-visibility set, Hamming graphs, Turán-type problem |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2025 |
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| Year of publishing: | 2025 |
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| Number of pages: | str. 63-78 |
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| Numbering: | Vol. 45, no. 1 |
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| PID: | 20.500.12556/DiRROS-21117  |
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| UDC: | 519.17 |
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| ISSN on article: | 1232-9274 |
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| DOI: | 10.7494/OpMath.2025.45.1.63 |
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| COBISS.SI-ID: | 220652803 |
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| Note: |
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| Publication date in DiRROS: | 30.12.2024 |
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| Views: | 35 |
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| Downloads: | 13 |
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| Metadata: |    |
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