20.500.12556/DiRROS-21117 Total mutual-visibility in Hamming graphs 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. mutual-visibility set total mutual-visibility set Hamming graphs Turán-type problem množica vzajemne vidnosti množica celotne vzajemne vidnosti Hammingovi grafi problem Turánovega tipa true false true Angleški jezik Slovenski jezik Neznano 2024-12-30 12:35:17 2024-12-30 12:35:17 2024-12-31 03:34:56 0000-00-00 00:00:00 2025 0 0 str. 63-78 no. 1 Vol. 45 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-01-01 519.17 1232-9274 10.7494/OpMath.2025.45.1.63 220652803 opuscula_math_4505.pdf opuscula_math_4505.pdf 1 76C8DA0C8283EF377B0D2CDA5E2781EF 98712cc067e35e913fe44abec846c1cd7ffa7ee6813be8f1bed2b79adb45e5fe 4921f2a5-c6a2-11ef-ba3f-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=29937 https://www.opuscula.agh.edu.pl/om-vol45iss1art5 1 25c2c9e0-c6a2-11ef-ba3f-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=29936 Inštitut za matematiko, fiziko in mehaniko 0 0 0