20.500.12556/RUL-154513 Packings in bipartite prisms and hypercubes The $2$-packing number $\rho_2(G)$ of a graph $G$ is the cardinality of a largest $2$-packing of $G$ and the open packing number $\rho^{\rm o}(G)$ is the cardinality of a largest open packing of $G$, where an open packing (resp. $2$-packing) is a set of vertices in $G$ no two (closed) neighborhoods of which intersect. It is proved that if $G$ is bipartite, then $\rho^{\rm o}(G\Box K_2) = 2\rho_2(G)$. For hypercubes, the lower bounds $\rho_2(Q_n) \ge 2^{n - \lfloor \log n\rfloor -1}$ and $\rho^{\rm o}(Q_n) \ge 2^{n - \lfloor \log (n-1)\rfloor -1}$ are established. These findings are applied to injective colorings of hypercubes. In particular, it is demonstrated that $Q_9$ is the smallest hypercube which is not perfect injectively colorable. It is also proved that $\gamma_t(Q_{2^k}\times H) = 2^{2^k-k}\gamma_t(H)$, where $H$ is an arbitrary graph with no isolated vertices. 2-packing number open packing number bipartite prism hypercube injective coloring total domination number 2-pakirno število odprto pakirno število dvodelna prizma hiperkocke injektivno barvanje celotno dominacijsko število true false true Angleški jezik Slovenski jezik Članek v reviji 2024-02-19 14:32:32 2024-02-19 14:32:35 2024-05-24 12:44:01 0000-00-00 00:00:00 2024 0 0 6 str. iss. 4, art. 113875 Vol. 347 Apr. 2024 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 0012-365X 10.1016/j.disc.2024.113875 181387523 1-s2.0-S0012365X24000062-main-2.pdf 1-s2.0-S0012365X24000062-main-2.pdf 1 BC4297894DBBF84BEE1923433A1D0D53 d87c4c783c33436d7e7c93199b060c1006c0c340737a836f16b04b1f2094b2b3 8947e052-cf2c-11ee-9559-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=180986 https://www.sciencedirect.com/science/article/pii/S0012365X24000062 1 58bae06d-cf2b-11ee-9559-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=180985 Fakulteta za matematiko in fiziko 0 0 0