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 $\gamma (G\mathrm{\square }H)\ge \frac{\pi (G)}{2\pi (G)-1}\gamma (G)\gamma (H)$. Our methods allow us to prove the following statements for any graphs $G$ and $H$, (1) $\gamma (G\mathrm{\square }H)\ge \frac{\lceil \frac{\gamma (G)}{2}\rceil }{2\lceil \frac{\gamma (G)}{2}\rceil -1}\gamma (G)\gamma (H)$ for odd $\gamma (G)$, (2) $\gamma (G\mathrm{\square }H)\ge \frac{\gamma (G)}{2\gamma (G)-2}\gamma (G)\gamma (H)$, for even $\gamma (G)$, and (3) a short proof of Vizing’s conjecture where $\gamma (G)=3$. Our argument relies on establishing efficient correspondences between dominating vertices and subsets of their neighborhoods and then showing a sufficient number of dominating vertices that horizontally dominate vertically undominated cells.