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1.
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The strong isometric dimension of graphs of diameter two
Janja Jerebic, Sandi Klavžar, 2003

Abstract: Krepka izometrična dimenzija ▫$textrm{idim}(G)$▫ grafa ▫$G$▫ je najmanjše število ▫$k$▫, za katero lahko ▫$G$▫ izometrično vložimo v krepki produkt ▫$k$▫ poti. Problem določitve ▫$textrm{idim}(G)$▫ za grafe premera dva je reduciran na problem pokrivanja komplementa grafa ▫$G$▫ s polnimi dvodelnimi grafi. Za primer je pokazano, da je izometrična dimenzija Petersenovega grafa enaka 5.
Keywords: matematika, teorija grafov, izometrični podgraf, krepki produkt grafov, premer grafa, krepka izometrična dimenzija, Petersenov graf, mathematics, graph theory, isometric subgraph, strong product of graphs, graph diameter, strong isometric dimension, Petersen graph
Published: 10.07.2015; Views: 464; Downloads: 19
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5.
Chromatic numbers of strong product of odd cycles
Janez Žerovnik, 2002, published scientific conference contribution

Abstract: The problem of determining the chromatic numbers of the strong product of cycles is considered. A construction is given proving ▫$chi(G) = 2^p + 1$▫ for a product of ▫$p$▫ odd cycles of lengths at least ▫$2^p + 1$▫. Several consequences are discussed. In particular it is proved that the strong product of ▫$p$▫ factors has chromatic number at most ▫$2^p + 1$▫ provided that each factor admits the homomorphism to sufficiently long odd cycle ▫$C_{m_i}, ; m_i ge 2^p + 1$▫.
Keywords: matematika, teorija grafov, krepki produkt grafov, kromatično število, lih cikel, minimalna neodvisna dominantna množica, mathematics, graph theory, strong product, chromatic number, odd cycle, minimal independent dominating set
Published: 10.07.2015; Views: 521; Downloads: 48
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6.
On the weak reconstruction of strong product graphs
Blaž Zmazek, Janez Žerovnik, 2003, published scientific conference contribution

Abstract: We prove that any nontrivial connected strong product graph can be uniquely reconstructed from each of its one vertex deleted subgraphs.
Keywords: matematika, teorija grafov, problem rekonstrukcije, šibka rekonstrukcija, krepki produkt grafov, mathematics, graph theory, reconstruction problem, strong product, composite graphs
Published: 10.07.2015; Views: 343; Downloads: 46
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7.
On induced and isometric embeddings of graphs into the strong product of paths
Janja Jerebic, Sandi Klavžar, 2006, original scientific article

Abstract: Primerjani sta krepka izometrična dimenzija in sosedna izometrična dimenzija grafov. Koncepta sta ekvivalentna za grafe premera 2 in v tem primeru se problem določitve dimenzije reducira na problem pokritja s polnimi dvodelnimi grafi. S pomočjo tega pristopa je določena krepka izometrična dimenzija in sosedna izometrična dimenzija za različne grafe (na primer za Petersenov graf). Podan je pozitiven odgovor na Problem 4.1 iz [Fitzpatrick, Nowakowski, The strong isometric dimension of finite reflexive graphs, Discuss. Math. Graph Theory 20 (2000) 23-38], ali obstaja tak graf ▫$G$▫, ki ima krepko izometrično dimenzijo večjo od ▫$lceil |V(G)|/2 rceil$▫.
Keywords: matematika, teorija grafov, krepki produkt grafov, krepka izometrična dimenzija, sosedna izometrična dimenzija, mathematics, graph theory, strong product of graphs, adjacent isometric dimension, strong isometric dimension
Published: 10.07.2015; Views: 432; Downloads: 55
URL Link to full text

8.
Weak reconstruction of strong product graphs
Blaž Zmazek, Janez Žerovnik, 2007, original scientific article

Abstract: We prove that any nontrivial connected strong product graph can be uniquely reconstructed from each of its one vertex deleted subgraphs.
Keywords: matematika, teorija grafov, problem rekonstrukcije, šibka rekonstrukcija, krepki produkt grafov, mathematics, graph theory, reconstruction problem, strong product, composite graphs
Published: 10.07.2015; Views: 334; Downloads: 61
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9.
Strong isometric dimension, biclique coverings, and Sperner's theorem
Dalibor Fronček, Janja Jerebic, Sandi Klavžar, Petr Kovář, 2007, original scientific article

Abstract: Krepka izometrična dimenzija grafa ▫$G$▫ je najmanjše število ▫$k$▫, tako da lahko ▫$G$▫ izometrično vložimo v krepki produkt ▫$k$▫ poti. Z uporabo Spernerjevega izreka je določena krepka izometrična dimenzija Hammingovih grafov ▫$K_2,{square}, K_n$▫.
Keywords: matematika, teorija grafov, krepka izometrična dimenzija, Hammingovi grafi, mathematics, graf theory, strong product, Hamming graphs
Published: 10.07.2015; Views: 375; Downloads: 41
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10.
NZ-flows in strong products of graphs
Wilfried Imrich, Iztok Peterin, Simon Špacapan, Cun-Quan Zhang, 2010, original scientific article

Abstract: Za krepki produkt ▫$G_1 boxtimes G_2$▫ grafov ▫$G_1$▫ in ▫$G_2$▫ dokažemo, da je ▫${mathbb{Z}}_3$▫-pretočno kontraktibilen natanko tedaj, ko ▫$G_1 boxtimes G_2$▫ ni izomorfen ▫$Tboxtimes K_2$▫ (kar poimenujemo ▫$K_4$▫-drevo), kjer je ▫$T$▫ drevo. Sledi, da za ▫$G_1 boxtimes G_2$▫ obstaja NZ 3-pretok, razen če je ▫$G_1 boxtimes G_2$▫ ▫$K_4$▫-drevo. Dokaz je konstruktiven in implicira polinomski algoritem, ki nam vrne NZ 3-pretok, če ▫$G_1 boxtimes G_2$▫ ni ▫$K_4$▫-drevo, oziroma NZ 4-pretok sicer.
Keywords: matematika, teorija grafov, celoštevilski pretoki, krepki produkt, poti, cikli, nikjer ničelni pretok, mathematics, graph theory, integer flows, strong product, paths, cycles
Published: 10.07.2015; Views: 359; Downloads: 59
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