375 KB43 KB200820252025Brešar, BoštjanCornet, María GraciaDravec, TanjaHenning, Michael A.16 str.letnik:25številka:4, article p4.02DOI:10.26493/1855-3974.3294.7fdISSN:1855-3966COBISSID_HOST:243733251URN:URN:NBN:SI:doc-T4VC5H4KenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporanea2-packing2-pakiranjek-dominacijak-dominationKneser graphsKneserjev grafk-terna celotna dominacijak-tuple total dominationk-domination invariants on Kneser graphs|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 graphsČlanek je nadaljevanje članka M.G. Cornet, P. Torres, arXiv:2308.15603, v katerem sta avtorja raziskovala ?$k$?-terno dominacijsko število in ?$2$?-pakirno število Kneserjevih grafov ?$K(n,r)$?. Nas zanimata dve sorodni inačici, namreč ?$k$?-dominacijsko število ?$\gamma_k(K(n,r))$? in ?$k$?-terno celotno dominacijsko število ?$\gamma_{t\times k}(K(n,r))$? Kneserjevih grafov ?$K(n,r)$?. Za obe invarianti dokažemo neko vrsto monotonosti in sicer, da za vse ?$n\ge 2(k+r)$? velja ?$\gamma_k(K(n,r))\ge \gamma_k(K(n+1,r))$? ter da za vse ?$n\ge 2r+1$? velja ?$\gamma_{t\times k}(K(n,r))\ge \gamma_{t\times k}(K(n+1,r))$?. Dokažemo tudi, da velja ?$\gamma_k(K(n,r))=\gamma_{t\times k}(K(n,r))=k+r$?, če je ?$n\geq r(k+r)$?, in da je v tem primerih vsaka ?$\gamma_k(K(n,r))$?-množica in ?$\gamma_{t\times k}$?-množica klika. Po drugi strani dokažemo, da velja ?$\gamma_k(r(k+r)-1,r)=\gamma_{t\times k}(r(k+r)-1,r)=k+r+1$? za vsak ?$k\ge 2$?. Glede ?$2$?-pakirnega števila ?$\rho_2(K(n,r))$? grafa ?$K(n,r)$? določimo točne vrednosti ?$\rho_2(K(3r-3,r))$?, ko je ?$r\ge 10$? in podamo zadostne pogoje v obliki mej za število ?$r$? glede na število ?$n$?, ki zagotavljajo, da je ?$\rho_2(K(n,r))$? enako nekim predpisanim majhnim vrednostim. Dokažemo tudi neko vrsto monotonosti za ?$2$?-pakirno število Kneserjevih grafovTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije