20.500.12556/DiRROS-23096 $k$-domination invariants on Kneser graphs k-dominacijske invariante v Kneserjevih grafih In this follow-up to work of M.G. Cornet and P. Torres from 2023, where the $k$-tuple domination number and the $2$-packing number in Kneser graphs $K(n,r)$ were studied, we are concerned with two variations, the $k$-domination number, $\gamma_k(K(n,r))$, and the $k$-tuple total domination number, $\gamma_{t\times k}(K(n,r))$, of $K(n,r)$. For both invariants we prove monotonicity results by showing that $\gamma_k(K(n,r))\ge \gamma_k(K(n+1,r))$ holds for any $n\ge 2(k+r)$, and $\gamma_{t\times k}(K(n,r))\ge \gamma_{t\times k}(K(n+1,r))$ holds for any $n\ge 2r+1$. We prove that $\gamma_k(K(n,r))= \gamma_{t\times k}(K(n,r))= k+r$ when $n\geq r(k+r)$, and that in this case every $\gamma_k$-set and $\gamma_{t\times k}$-set is a clique, while $\gamma_k(r(k+r)-1,r)=\gamma_{t\times k}(r(k+r)-1,r)=k+r+1$, for any $k\ge 2$. Concerning the $2$-packing number, $\rho_2(K(n,r))$, of $K(n,r)$, we prove the exact values of $\rho_2(K(3r-3,r))$ when $r\ge 10$, and give sufficient conditions for $\rho_2(K(n,r))$ to be equal to some small values by imposing bounds on $r$ with respect to $n$. We also prove a version of monotonicity for the $2$-packing number of Kneser graphs. Kneser graphs k-domination k-tuple total domination 2-packing Kneserjev graf k-dominacija k-terna celotna dominacija 2-pakiranje true false true Angleški jezik Slovenski jezik Neznano 2025-07-24 13:34:10 2025-07-24 13:34:11 2025-10-24 03:53:09 0000-00-00 00:00:00 2025 0 0 16 str. no. 4, [article no.] P4.02 Vol. 25 2025 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2025-01-01 519.17 1855-3966 10.26493/1855-3974.3294.7fd 243733251 amc_3294.pdf amc_3294.pdf 1 66EC8CE06F64C6DE4D26867E1D16C882 4e62f19c819249b8e6b673fc3043ab61dfacd93a9690898a7a26f49df6a21e78 be59013f-6882-11f0-9bb7-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=33854 https://amc-journal.eu/index.php/amc/article/view/3294 1 206fae35-6882-11f0-9bb7-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=33853 Inštitut za matematiko, fiziko in mehaniko 0 0 0