An improved constant for Vizing's conjecture

For any graph $G = (V,E)$, a subset $S {\subseteq} V$ dominates $G$ if $N[S] = V$. The minimum cardinality over all such $S$ is called the domination number, written ${\gamma}(G)$. The classical conjecture of V.G. Vizing states that ${\gamma}(G{\square} H) {\ge} {\gamma}(G){\gamma}(H)$ where ${\square}$ stands for the Cartesian product of graphs. In this paper, we apply well-known results to prove the Vizing-type inequality ${\gamma}(G{\square} H) {\ge} .5809 {\gamma}(G){\gamma}(H)$.