Two inequalities related to Vizing's conjecture

Shira Zerbib

Published 2017-06-12Version 1

A well-known conjecture of Vizing is that $\gamma(G \square H) \ge \gamma(G)\gamma(H)$ for any pair of graphs $G, H$, where $\gamma$ is the domination number and $G \square H$ is the Cartesian product of $G$ and $H$. Suen and Tarr, improving a result of Clark and Suen, showed $\gamma(G \square H) \ge \frac{1}{2}\gamma(G)\gamma(H) + \frac{1}{2}\min(\gamma(G),\gamma(H))$. We further improve their result by showing $\gamma(G \square H) \ge \frac{1}{2}\gamma(G)\gamma(H) + \frac{1}{2}\max(\gamma(G),\gamma(H)).$ We also prove a fractional version of Vizing's conjecture: $\gamma(G \square H) \ge \gamma(G)\gamma^*(H)$.