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A $$\frac{3}{4}$$-approximation of Vizing’s conjecture for claw-free graphs. (English) Zbl 1443.05138
Summary: Vizing’s conjecture [V. G. Vizing, Vychisl. Sistemy, Novosibirsk 9, 30–43 (1963; Zbl 0194.25203)] 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 $$\gamma (G \square H) \geq \frac{3}{4} \gamma (G) \gamma (H)$$ always holds.
##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C76 Graph operations (line graphs, products, etc.)
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