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On construction for trees making the equality hold in Vizing's conjecture
Journal of Graph Theory, 2022AbstractIn 1968, Vizing gave a conjecture: for any graphs and , which is now still open. In the text book “Domination in Graph: Advanced Topices” edited by Haynes et al., they listed such a question: is there a structural characterization of the graphs such that there exists a graph with ?
Weisheng Zhao 0002 +2 more
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Vizing's conjecture: a survey and recent results
Journal of Graph Theory, 2011AbstractVizing'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.
Bostjan Bresar +6 more
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A 34-approximation of Vizing’s conjecture for claw-free graphs
Discrete Applied Mathematics, 2020Abstract 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. We prove that for any claw-free graph G and an arbitrary graph H , the inequality γ ( G □ H ) ≥ 3 4 γ ( G ) γ ( H ) always ...
Bostjan Bresar, Michael A. Henning
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Some improved inequalities related to Vizing's conjecture
Information Processing Letters, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Li-Dan Pei, Xiang-Feng Pan, Fu-Tao Hu
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Reducing Vizing’s 2-Factor Conjecture to Meredith Extension of Critical Graphs
Graphs and Combinatorics, 2020zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Xiaodong Chen, Qing Ji, Mingda Liu
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Domination in graphs: Vizing's conjecture
2022Vizing's conjecture remains one of the biggest open problems in domination in graph theory today. The conjecture states that the domination number of the Cartesian product of two graphs is at least as large as the product of the domination numbers of the two factor graphs.
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Application of upper and lower bounds for the domination number to Vizing's conjecture
Ars Comb., 2003If \(G_1\), \(G_2\) are graphs with the vertex sets \(V_1\), \(V_2\), then \(G_1 \square \, G_2\) is the graph whose vertex set is \(V_1 \times V_2\) and in which two vertices \((x_1, x_2)\), \((y_1, y_2)\) are adjacent if and only if either \(x_1, y_1\) are adjacent in \(G_1\) and \(x_2 = y_2\), or \(x_1 = y_1\) and \(x_2, y_2\) are adjacent in \(G_2\)
William Edwin Clark +2 more
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