An Improved Inequality Related to Vizing's Conjecture

Abstract

Vizing conjectured in 1963 that $\gamma(G \Box H) \geq \gamma(G)\gamma(H)$ for any graphs $G$ and $H$. A graph $G$ is said to satisfy Vizing's conjecture if the conjectured inequality holds for $G$ and any graph $H$.  Vizing's conjecture has been proved for $\gamma(G) \le 3$, and it is known to hold for other classes of graphs. Clark and Suen in 2000 showed that $\gamma(G \Box H) \geq \frac{1}{2}\gamma(G)\gamma(H)$ for any graphs $G$ and $H$.   We give a slight improvement of this inequality by tightening their arguments.