1. A new framework to approach Vizing's conjectureBoštjan Brešar, Bert L. Hartnell, Michael A. Henning, Kirsti Kuenzel, Douglas F. Rall, 2021, original scientific article Abstract: We introduce a new setting for dealing with the problem of the domination number of the Cartesian product of graphs related to Vizing's conjecture. The new framework unifies two different approaches to the conjecture. The most common approach restricts one of the factors of the product to some class of graphs and proves the inequality of the conjecture then holds when the other factor is any graph. The other approach utilizes the so-called Clark-Suen partition for proving a weaker inequality that holds for all pairs of graphs. We demonstrate the strength of our framework by improving the bound of Clark and Suen as follows: ɣ(X◻Y) ≥ max{1/2ɣ(X) ɣt(Y), 1/2ɣt(X) ɣ(Y)}, where ɣ stands for the domination number, ɣt is the total domination number, and X◻Y is the Cartesian product of graphs X and Y.
Keywords: Cartesian product, total domination, Vizing's conjecture, Clark and Suen bound
Published in DKUM: 09.08.2024; Views: 86; Downloads: 7
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2. Vizing's conjecture: a survey and recent resultsBoštjan Brešar, Paul Dorbec, Wayne Goddard, Bert L. Hartnell, Michael A. Henning, Sandi Klavžar, Douglas F. Rall, 2012, review article Abstract: Vizingova domneva iz leta 1968 trdi, da je dominacijsko število kartezičnega produkta dveh grafov vsaj tako veliko, kot je produkt dominacijskih števil faktorjev. V članku naredimo pregled različnih pristopov k tej osrednji domnevi iz teorije grafovske dominacije. Ob tem dokažemo tudi nekaj novih rezultatov. Tako so na primer pokazane nove lastnosti minimalnega protiprimera, dokazana je tudi nova spodnja meja za produkte grafov brez induciranega ▫$K_{1,3}$▫ s poljubnimi grafi. Skozi celoten članek so obravnavani pripadajoči odprti problemi, vprašanja in sorodne domneve.
Keywords: matematika, teorija grafov, kartezični produkt, dominacija, Vizingova domneva, mathematics, graph theory, Caretesian product, domination, Vizing's conjecture
Published in DKUM: 10.07.2015; Views: 1362; Downloads: 90
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3. Vizing's conjecture: a survey and recent resultsBoštjan Brešar, Paul Dorbec, Wayne Goddard, Bert L. Hartnell, Michael A. Henning, Sandi Klavžar, Douglas F. Rall, 2009 Abstract: Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this paper we survey the approaches to this central conjecture from domination theory and give some new results along the way. For instance, several new properties of a minimal counterexample to the conjecture are obtained and a lower bound for the domination number is proved for products of claw-free graphs with arbitrary graphs. Open problems, questions and related conjectures are discussed throughout the paper.
Keywords: matematika, teorija grafov, kartezični produkt, dominacija, Vizingova domneva, mathematics, graph theory, Caretesian product, domination, Vizing's conjecture
Published in DKUM: 10.07.2015; Views: 1365; Downloads: 99
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4. Fair reception and Vizing's conjectureBoštjan Brešar, Douglas F. Rall, 2009, original scientific article Abstract: Vpeljemo koncept poštenega sprejema grafa, ki je povezan z njegovim dominantnim številom. Dokažemo, da za vse grafe, ki imajo pošten sprejem velikosti njihovega dominantnega števila, velja Vizingova domneva o dominantnem številu kartezičnega produkta grafov, s čimer posplošimo dobro znan rezultat Barcalkina in Germana o razstavljivih grafih. S kombiniranjem nav sega koncepta in rezultata Aharonija, Bergerja in Ziva dobimo alternativen dokaz izreka Aharonija in Szaba, ki pravi, da tetivni grafi zadoščajo Vizingovi domnevi. Predstavimo tudi novo neskončno družino grafov, ki zadoščajo Vizingovi domnevi.
Keywords: matematika, teorija grafov, dominacija, kartezični produkt grafov, Vizingova domneva, mathematics, graph theory, domination, Cartesian product of graphs, Vizing's conjecture
Published in DKUM: 10.07.2015; Views: 1254; Downloads: 113
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5. Domination gameBoštjan Brešar, Sandi Klavžar, Douglas F. Rall, 2009 Abstract: The domination game played on a graph ▫$G$▫ consists of two players, Dominator and Staller who alternate taking turns choosing a vertex from ▫$G$▫ such that whenever a vertex is chosen the graph in as few steps as possible and Staller wishes to delay the process as much as possible. The game domination number ▫$gamma_g(G)$▫ is the number of vertices chosen when Dominator starts the game and the Staller-start game domination number ▫$gamma'_g(G)$▫ when Staller starts the game. It is proved that for any graph ▫$G$▫, ▫$gamma(G) le gamma_g(G) le 2gamma(G) - 1$▫, and that all possible values can be realized. It is also proved that for any graph ▫$G$▫, ▫$gamma_g(G) - 1 le gamma'_g(G) le gamma_g(G) + 2$▫, and that most of the possibilities for mutual values of ▫$gamma_g(G)$▫ and ▫$gamma'_g(G)$▫ can be realized. A connection with Vizing's conjecture is established and several problems and conjectures stated.
Keywords: teorija grafov, teorija iger, dominantnost, Vizingova domneva, graph theory, game theory, domination, domination game, game domination number, Vizing's conjecture
Published in DKUM: 10.07.2015; Views: 1963; Downloads: 31
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6. On integer domination in graphs and Vizing-like problemsBoštjan Brešar, Michael A. Henning, Sandi Klavžar, 2006, original scientific article Abstract: Nadaljujemo študij ▫${k}$▫-dominantnih funkcij v grafih (ali, kot bomo tudi rekli, celoštevilske dominacije), ki so jo začeli Domke, Hedetniemi, Laskar in Fricke. Za celo število ▫$k ge 1$▫ je funkcija ▫$f: V(G) to {0,1,...,k}$▫, definirana na točkah grafa ▫$G$▫, ▫${k}$▫-dominantna funkcija, če je vsota funkcijskih vrednosti na vsaki zaprti okolici vsaj ▫$k$▫. Teža ▫${k}$▫-dominantne funkcije je vsota funkcijskih vrednosti po vseh točkah. ▫${k}$▫-dominantno število grafa ▫$G$▫ je najmanjša teža ▫${k}$▫-dominantne funkcije na ▫$G$▫. Obravnavamo ▫${k}$▫-dominantno število kartezičnega produkta grafov, predvsem probleme povezane s slavno Vizingovo domnevo. Študirana je tudi povezava med ▫${k}$▫-dominantnim številom in drugimi tipi dominacijskih parametrov.
Keywords: matematika, teorija grafov, ▫${k}$▫-dominantna funkcija, celoštevilska dominacija, Vizingova domneva, kartezični produkt grafov, mathematics, graph theory, ▫${k}$▫-dominating function, integer domination, Vizing's conjecture, Cartesian product
Published in DKUM: 10.07.2015; Views: 1280; Downloads: 68
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7. Behzad-Vizing conjecture and Cartesian-product graphsBlaž Zmazek, Janez Žerovnik, 2004, published scientific conference contribution Abstract: We prove the following theorem: if the Behzad-Vizing conjecture is true for graphs ▫$G$▫ and ▫$H$▫, then is it true for the cartesian product ▫$G Box H$▫.
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 in DKUM: 10.07.2015; Views: 1548; Downloads: 87
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