A new bound for Vizing's conjecture
  For any graph $G$, we define the power $\pi(G)$ as the minimum of the largest number of neighbors in a $\gamma$-set of $G$, of any vertex, taken over all $\gamma$-sets of $G$. We show that $\gamma(G\square H)\geq \frac{\pi(G)}{2\pi(G) -1}\gamma(G)\gamma(H)$. This implies that for any graphs $G$ and $H$, $\gamma(G\square H)\geq \frac{\gamma(G)}{2\gamma(G)-1}\gamma(G)\gamma(H)$, and if $G$ is claw-free or $P_4$-free, $\gamma(G\square H)\geq \frac{2}{3}\gamma(G)\gamma(H)$, where $\gamma(G)$ is the domination number of $G$. 
Elliot Krop
math.CO