A new bound for Vizing's conjecture

Abstract

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 $\gamma(G\square H)\geq \frac{\pi(G)}{2\pi(G) -1}\gamma(G)\gamma(H)$. Our methods allow us to prove the following statements for any graphs $G$ and $H$, (1) $\gamma(G\square H)\geq \frac{\lceil \frac{\gamma (G)}{2}\rceil}{2\lceil \frac{\gamma (G)}{2}\rceil-1}\gamma(G)\gamma(H)$ for odd $\gamma(G)$, (2) $\gamma(G\square H)\geq \frac{\gamma (G)}{2\gamma (G)-2}\gamma(G)\gamma(H)$, 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.