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 $\gamma (G\mathrm{\square }H)\ge \gamma (G)\gamma (H)$ where $\mathrm{\square }$ stands for the Cartesian product of graphs. In this paper, we apply well-known results to prove the Vizing-type inequality $\gamma (G\mathrm{\square }H)\ge .5809\gamma (G)\gamma (H)$.