An Improvement on Vizing's Conjecture

Abstract

Let $\gamma(G)$ denote the domination number of a graph $G$. A {\it Roman domination function} of a graph $G$ is a function $f: V\to\{0,1,2\}$ such that every vertex with 0 has a neighbor with 2. The {\it Roman domination number} $\gamma_R(G)$ is the minimum of $f(V(G))=\Sigma_{v\in V}f(v)$ over all such functions. Let $G\square H$ denote the Cartesian product of graphs $G$ and $H$. We prove that $\gamma(G)\gamma(H) \leq \gamma_R(G\square H)$ for all simple graphs $G$ and $H$, which is an improvement of $\gamma(G)\gamma(H) \leq 2\gamma(G\square H)$ given by Clark and Suen \cite{CS}, since $\gamma(G\square H)\leq \gamma_R(G\square H)\leq 2\gamma(G\square H)$.