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1.
On the vital difference between number theory and numerology in physics
Leila Marek-Crnjac, 2008, original scientific article

Abstract: The interrelation between pure mathematics and physics is very deep and intricate. It is similar to the duality between pure reason and reality. However, the difference between number theory and numerology when used in physics should be clear and easily sorted out. Confusion in this respect is not possible and while numerology must be deplored, number theoretical argument should be an integral part of theoretical physics.
Keywords: number theory, topology, numerology
Published: 31.05.2012; Views: 782; Downloads: 58
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The b-chromatic number of cubic graphs
Marko Jakovac, Sandi Klavžar, 2009, published scientific conference contribution abstract

Keywords: graph theory, chromatic number, graphs, Petersen graph, cubic graphs
Published: 07.06.2012; Views: 1003; Downloads: 85
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4.
On total chromatic number of direct product graphs
Katja Prnaver, Blaž Zmazek, 2009, original scientific article

Keywords: graph theory, total chromatic number, direct product, tensor product
Published: 07.06.2012; Views: 1116; Downloads: 48
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5.
On domination numbers of graph bundles
Blaž Zmazek, Janez Žerovnik, 2005

Abstract: Let ▫$gamma(G)$▫ be the domination number of a graph ▫$G$▫. It is shown that for any ▫$k ge 0$▫ there exists a Cartesian graph bundle ▫$B Box_varphi F$▫ such that ▫$gamma(B Box_varphi F) = gamma(B)gamma(F) - 2k$▫. The domination numbers of Cartesian bundles of two cycles are determined exactly when the fibre graph is a triangle or a square. A statement similar to Vizing's conjecture on strong graph bundles is shown not to be true by proving the inequality ▫$gamma(B boxtimes_varphi F) le gamma(B)gamma(F)$▫ for strong graph bundles. Examples of graphs ▫$B$▫ and ▫$F$▫ with ▫$gamma(B boxtimes_varphi F) < gamma(B)gamma(F)$▫ are given.
Keywords: matematika, teorija grafov, kartezični produkt grafov, dominantno število, dominantna množica, grafovski sveženj, mathematics, graph theory, graph bundle, dominating set, domination number, Cartesian product
Published: 10.07.2015; Views: 491; Downloads: 53
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6.
The minor crossing number of graphs with an excluded minor
Drago Bokal, Gašper Fijavž, David Richard Wood, 2008, original scientific article

Abstract: The minor crossing number of a graph ▫$G$▫ is the minimum crossing number of a graph that contains ▫$G$▫ as a minor. It is proved that for every graph ▫$H$▫ there is a constant ▫$c$▫, such that every graph ▫$G$▫ with no ▫$H$▫-minor has minor crossing number at most ▫$c|V(G)|$▫.
Keywords: mathematics, graph theory, graph minor, excluded minor, crossing number, minor crossing number
Published: 10.07.2015; Views: 271; Downloads: 42
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7.
The distinguishing chromatic number of Cartesian products of two complete graphs
Janja Jerebic, Sandi Klavžar, 2008

Abstract: Označitev grafa ▫$G$▫ je razlikovalna, če jo ohranja le trivialni avtomorfizem grafa ▫$G$▫. Razlikovalno kromatično število grafa ▫$G$▫ je najmanjše naravno število, za katero obstaja razlikovalna označitev grafa, ki je hkrati tudi dobro barvanje. Za vse ▫$k$▫ in ▫$n$▫ je določeno razlikovalno kromatično število kartezičnih produktov ▫$K_kBox K_n$▫. V večini primerov je enako kromatičnemu številu, kar med drugim odgovori na vprašanje Choia, Hartkeja and Kaula, ali obstajajo še kakšni drugi grafi, za katere velja enakost.
Keywords: teorija grafov, razlikovalno kromatično število, grafovski avtomorfizem, kartezični produkt grafov, graph theory, distinguishing chromatic number, graph automorphism, Cartesian product of graphs
Published: 10.07.2015; Views: 345; Downloads: 41
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8.
On the sharpness of some results relating cuts and crossing numbers
Laurent Beaudou, Drago Bokal, 2010, original scientific article

Abstract: It is already known that for very small edge cuts in graphs, the crossing number of the graph is at least the sum of the crossing number of (slightly augmented) components resulting from the cut. Under stronger connectivity condition in each cut component that was formalized as a graph operation called zip product, a similar result was obtained for edge cuts of any size, and a natural question was asked, whether this stronger condition is necessary. In this paper, we prove that the relaxed condition is not sufficient when the size of the cut is at least four, and we prove that the gap can grow quadratically with the cut size.
Keywords: mathematics, graph theory, crossing number, zip product, graph cuts
Published: 10.07.2015; Views: 312; Downloads: 26
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9.
Domination game played on trees and spanning subgraphs
Boštjan Brešar, Sandi Klavžar, Douglas F. Rall, 2011

Abstract: Igra dominacije na grafu ▫$G$▫ je bila vpeljana v [B. Brešar, S. Klavžar, D. F. Rall, Domination game and an imagination strategy, SIAM J. Discrete Math. 24 (2010) 979-991]. Dva igralca, Dominator in Zavlačevalec, drug za drugim izbirata po eno vozlišče grafa. Vsako izbrano vozlišče mora povečati množico vozlišč, ki so bila dominirana do tega trenutka igre. Oba igralca izbirata optimalno strategijo, pri čemer Dominator želi igro končati v najmanjšem možnem številu korakov, Zavlačevalec pa v največjem možnem številu korakov. Igralno dominacijsko število ▫$gamma_g(G)$▫ je število izbranih vozlišč v igri, kjer je Dominator prvi izbral vozlišče. Ustrezno invarianto, ko igro začne Zavlačevalec, označimo z ▫$gamma_g'(G)$▫. V članku sta obe igri proučevani na drevesih in vpetih podgrafih. Dokazana je spodnja meja za igralno dominacijsko število drevesa, ki je funkcija njegovega reda in maksimalne stopnje. Pokazano je, da je meja asimptotično optimalna. Dokazano je, da za vsak ▫$k$▫ obstaja drevo ▫$T$▫ z ▫$(gamma_g(T),gamma_g'(T)) = (k,k+1)$▫ in postavljena je domneva, da ne obstaja drevo z ▫$(gamma_g(T),gamma_g'(T)) = (k,k-1)$▫. Obravnavana je povezava med igralnim dominacijskim številom grafa in njegovimi vpetimi podgrafi. Dokazano je, da za vsako naravno število ▫$ell geq 1$▫ obstaja graf ▫$G$▫ z vpetim drevesom ▫$T$▫, tako da velja ▫$gamma_g(G)-gamma_g(T)ge ell$▫. Nadalje obstajajo 3-povezani grafi ▫$G$▫, ki imajo vpeta drevesa z igralnim dominacijskim številom poljubno manjšim od ▫$G$▫.
Keywords: igra dominacije, igralno dominacijsko število, drevo, vpeti podgraf, graph theory, domination game, game domination number, tree, spanning subgraph
Published: 10.07.2015; Views: 602; Downloads: 47
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10.
On the k-path vertex cover of some graph products
Marko Jakovac, Andrej Taranenko, 2013, original scientific article

Abstract: A subset S of vertices of a graph G is called a k-path vertex cover if every path of order k in G contains at least one vertex from S. Denote by ▫$psi_k$▫(G) the minimum cardinality of a k-path vertex cover in G. In this paper, improved lower and upper bounds for ▫$psi_k$▫ of the Cartesian and the strong product of paths are derived. It is shown that for ▫$psi_3$▫ those bounds are tight. For the lexicographic product bounds are presented for ▫$psi_k$▫, moreover ▫$psi_2$▫ and ▫$psi_3$▫ are exactly determined for the lexicographic product of two arbitrary graphs. As a consequence the independence and the dissociation number of the lexicographic product are given.
Keywords: matematika, teorija grafov, vozliščno pokritje, po poteh vozliščno pokritje, disociacijsko število, neodvisnostno število, grafovski produkti, mathematics, graph theory, vertex cover, path vertex cover, dissociation number, independence number, graph products
Published: 10.07.2015; Views: 419; Downloads: 7
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