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Vizing conjectured that $\\gamma(G \\square H) \\geq \\gamma(G)\\gamma(H)$ where $\\square$ stands for the Cartesian product of graphs. In this note, we prove that if $\\left|G\\right|\\geq \\gamma(G)\\gamma(H)$ and $\\left|H\\right|\\geq \\gamma(G)\\gamma(H)$, then the conjecture holds. 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The minimum cardinality over all such $S$ is called the domination number, written $\\gamma(G)$. In 1963, V.G. Vizing conjectured that $\\gamma(G \\square H) \\geq \\gamma(G)\\gamma(H)$ where $\\square$ stands for the Cartesian product of graphs. In this note, we prove that if $\\left|G\\right|\\geq \\gamma(G)\\gamma(H)$ and $\\left|H\\right|\\geq \\gamma(G)\\gamma(H)$, then the conjecture holds. 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