Article
On a Vizinglike conjecture for direct product graphs
Department of Mathematics, PEF, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
Discrete Mathematics (Impact Factor: 0.56). 09/1996; 156(13):243246. DOI: 10.1016/0012365X(96)000325 Source: DBLP
ABSTRACT
Let fl(G) be the domination number of a graph G, and let G \Theta H be thedirect product of graphs G and H . It is shown that for any k 0 there existsa graph G such that fl(G \Theta G) fl(G)
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 "It has been shown that the Kronecker product is a good method to construct lager networks that can generate many good properties of the factor graphs (see [9]), and has received much research attention recently. Some properties and graphic parameters have been investigated [1] [2] [5] [8] [11]. The connectivity and diameter are two important parameters to measure reliability and efficiency of a network. "

 "A year later, Nowakowski and Rall [12] gave a counterexample for the latter conjecture. The same year, Klavžar and Zmazek [9] found an infinite series of counterexamples. Moreover, using these graphs they showed that the difference γ (G)γ (H) − γ (G × H) can be arbitrarily large. "
Article: Lower bounds for the domination number and the total domination number of direct product graphs
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ABSTRACT: A sharp lower bound for the domination number and the total domination number of the direct product of finitely many complete graphs is given: γ(×i=1tKni)≥t+1,t≥3. Sharpness is established in the case when the factors are large enough in comparison to the number of factors. The main result gives a lower bound for the domination (and the total domination) number of the direct product of two arbitrary graphs: γ(G×H)≥γ(G)+γ(H)−1γ(G×H)≥γ(G)+γ(H)−1. Infinite families of graphs that attain the bound are presented. For these graphs it also holds that γt(G×H)=γ(G)+γ(H)−1γt(G×H)=γ(G)+γ(H)−1. Some additional parallels with the total domination number are made. 
 "In the same paper an example was given that disproved a Vizingtype conjecture from [6]. An infinite series of such examples was presented in [12]. Chérifi, Gravier, Lagraula, Payan, and Zighem [3] determined the domination number of the direct product of two paths with * University of Maribor, FEECS, Smetanova 17, 2000 Maribor, Slovenia. "
Article: Dominating direct products of graphs
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ABSTRACT: An upper bound for the domination number of the direct product of graphs is proved. It in particular implies that for any graphs G and H, γ(G×H)⩽3γ(G)γ(H). Graphs with arbitrarily large domination numbers are constructed for which this bound is attained. Concerning the upper domination number we prove that Γ(G×H)⩾Γ(G)Γ(H), thus confirming a conjecture from [R. Nowakowski, D.F. Rall, Associative graph products and their independence, domination and coloring numbers, Discuss. Math. Graph Theory 16 (1996) 53–79]. Finally, for paireddomination of direct products we prove that γpr(G×H)⩽γpr(G)γpr(H) for arbitrary graphs G and H, and also present some infinite families of graphs that attain this bound.