388 KB35 KB200820252015Anholcer, MarcinCichacz, SylwiaPeterin, IztokTepeh, Aleksandraštevilka:1letnik:9str. 93-107COBISSID:17280857ISSN:1855-3966URN:URN:NBN:SI:doc-WYZI63ROenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaAbelian groupAbelova grupadirect product of graphsdirektni produkt grafovdistance magic labelinggroup labelinggrupno označevanjerazdaljno magično označevanjeGroup distance magic labeling of direct product of graphs|Let ?$G = (V,E)$? be a graph and ?$\Gamma$? an Abelian group, both of order ?$n$?. A group distance magic labeling of ?$G$? is a bijection ?$\ell \colon V \rightarrow \Gamma$? for which there exists ?$\mu \in \Gamma$? such that ?$\sum_{x \in N(v)}\ell (x) = \mu $? for all ?$v \in V$?, where ?$N(v)$? is the neighborhood of ?$v$?. In this paper we consider group distance magic labelings of direct product of graphs. We show that if ?$G$? is an ?$r$?-regular graph of order ?$n$? and ?$m = 4$? or ?$m = 8$? and ?$r$? is even, then the direct product ?$C_{m} \times G$? ?$\Gamma$?-distance magic for every Abelian group of order ?$mn$?. We also prove that ?$C_{m} \times C_{n}$? is ?$\mathbb{Z}_{mn}$?-distance magic if and only if ?$m \in \{4,8\}$? or ?$n \in \{4,8\}$? or ?$m,n \equiv 0 \pmod 4$?. It is also shown that if ?$m,n \not\equiv 0 \pmod 4$? then ?$C_{m} \times C_{n}$? is not ?$\Gamma$?-distance magic for any Abelian group ?$\Gamma$? of order ?$mn$?Naj bo ?$G = (V,E)$? graf in ?$\Gamma$? Abelova grupa, oba reda ?$n$?. Grupno razdaljno magično označevanje grafa ?$G$? je bijekcija ?$\ell \colon V \rightarrow \Gamma$?, za katero obstaja tak ?$\mu \in \Gamma$?, da je ?$\sum_{x\in N(v)}\ell (x) = \mu$? za vsak ?$v \in V$?, kjer je ?$N(v)$? soseščina vozlišča ?$v$?. V članku obravnavamo grupno razdaljno magično označevanje direktnih produktov grafov. Pokažemo, da če je ?$G$? ?$r$?-regularen graf reda ?$n$? in ?$m=4$? ali ?$m=8$? in je ?$r$? sodo število, potem je direktni produkt ?$C_{m} \times G$? ?$\Gamma$?-razdaljno magični graf za vsako Abelovo grupo reda ?$mn$?. Pokažemo tudi, da je ?$C_{m} \times C_{n}$? ?$\mathbb{Z}_{mn}$?-razdaljno magični graf natanko tedaj, ko je ?$m \in \{4,8\}$? ali ?$n \in \{4,8\}$? ali ?$m,n \equiv 0 \pmod 4$?. Prav tako pokažemo, da če ?$m,n \not\equiv 0 \pmod 4$?, potem ?$C_{m}\times C_{n}$? ni ?$\Gamma$?-razdaljno magični graf za nobeno Abelovo grupo ?$\Gamma$? reda ?$mn$?TEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije