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An improved bound in Vizing’s conjecture. (English) Zbl 1431.05118
Summary: A well-known conjecture of V. G. Vizing [Vychisl. Sist., Novosibirsk 9, 30–43 (1963; Zbl 0194.25203)] is that $$\gamma (G \Box H) \ge \gamma(G) \gamma(H)$$ for any pair of graphs $$G, H$$, where $$\gamma$$ is the domination number and $$G \Box H$$ is the Cartesian product of $$G$$ and $$H$$. S. Suen and J. Tarr [Electron. J. Comb. 19, No. 1, Research Paper P8, 4 p. (2012; Zbl 1243.05190)] improving a result of S. Suen and J. Tarr [Electron. J. Comb. 19, No. 1, Research Paper P8, 4 p. (2012; Zbl 1243.05190)], showed $$\gamma (G \Box H) \ge \frac{1}{2}\gamma (G)\gamma (H) + \frac{1}{2}\min(\gamma (G), \gamma (H))$$. We further improve their result by showing $$\gamma (G \Box H) \ge \frac{1}{2}\gamma(G) \gamma(H) + \frac{1}{2}\max(\gamma(G), \gamma(H))$$.

MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 05C76 Graph operations (line graphs, products, etc.)
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References:
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