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1.
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: 1189; Downloads: 110
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2.
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: 1299; Downloads: 63
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3.
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: 474; Downloads: 57
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On Groebner bases and their use in solving some practical problems
Matej Mencinger, 2013, original scientific article

Abstract: Groebner basis are an important theoretical building block of modern (polynomial) ring theory. The origin of Groebner basis theory goes back to solving some theoretical problems concerning the ideals in polynomial rings, as well as solving polynomial systems of equations. In this article four practical applications of Groebner basis theory are considered; we use Groebner basis to solve the systems of nonlinear polynomial equations, to solve an integer programming problem, to solve the problem of chromatic number of a graph, and finally we consider an original example from the theory of systems of ordinary (polynomial) differential equations. For practical computations we use systems MATHEMATICA and SINGULAR .
Keywords: polynomial system of (differential) equations, integer linear programming, chromatic number of a graph, polynomial rings, Groebner basis, CAS systems
Published: 10.07.2015; Views: 511; Downloads: 30
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6.
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: 671; Downloads: 54
URL Link to full text

7.
Behzad-Vizing conjecture and Cartesian-product graphs
Blaž Zmazek, Janez Žerovnik, 2004, published scientific conference contribution

Abstract: Dokazali smo naslednji izrek: Če Behzad-Vizingova domneva velja za grafa ▫$G$▫ in ▫$H$▫, potem velja tudi za kartezični produkt ▫$G Box H$▫.
Keywords: matematika, teorija grafov, kartezični produkt grafov, kromatično število, popolno kromatično število, Vizingova domneva, mathematics, graph theory, Cartesian graph product, chromatic number, total chromatic number, Vizing conjecture
Published: 10.07.2015; Views: 664; Downloads: 60
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Nonrepetitive colorings of trees
Boštjan Brešar, J. Grytczuk, Sandi Klavžar, S. Niwczyk, Iztok Peterin, 2007, original scientific article

Abstract: Barvanje vozlišč grafa ▫$G$▫ je neponavljajoče, če nobena pot v ▫$G$▫ ne tvori zaporedja sestavljenega iz dveh identičnih blokov. Najmanjše število barv, ki jih potrebujemo za tako barvanje, je Thuejevo kromatično število, označimo ga s ▫$pi(G)$▫. Slavni Thuejev izrek trdi, da je ▫$pi(P) = 3$▫ za vsako pot ▫$P$▫ z vsaj štirimi vozlišči. V članku študiramo Thuejevo kromatično število na drevesih. Glede na to,da je v tem razredu ▫$pi(T)$▫ omejeno s 4, je naš namen opisati 4-kromatična drevesa. V posebnem obravnavamo 4-kritična drevesa, ki so minimalna glede na to lastnost. Čeprav obstaja mnogo dreves ▫$T$▫ s ▫$pi(T) = 4$▫, pokažemo, da ima vsako od njih primerno veliko subdivizijo ▫$H$▫, tako da je ▫$pi(H)=3$▫. Dokaz se opira na Thuejeva zaporedja z dodatnimi lastnostmi, ki vključujejo palindromske besede. Obravnavamo tudi neponavljajoča barvanja povezav na drevesih. S podobnimi argumenti dokažemo, da ima vsako drevo subdivizijo, ki jo lahko po povezavah pobarvamo z največ ▫$Delta +1$▫ barvami brez ponavljanja na poteh.
Keywords: kombinatorika na besedah, neponavljajoče zaporedje, Thuejevo kromatično število, drevo, palindrom, combinatorics on words, nonrepetitive sequence, Thue chromatic number, tree, palindrome
Published: 10.07.2015; Views: 540; Downloads: 67
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