On domination of Cartesian product of directed cycles

Abstract

Let $\gamma (C_m\square C_n)$ be the domination number of the Cartesian product of directed cycles $C_m$ and $C_n$ for $m,n\ge 2$. Shaheen [] and Liu and al.[ ], [ ] determined the value of $\gamma (C_m\square C_n)$ when $m\le 6$ and when both $m$ and $n$ $\equiv 0$ $(mod3)$. In this article we give, in general, the value of $\gamma (C_m\square C_n)$ when $m\equiv 2$ $(mod3)$ and improve the known lower bound for most of the remaining cases. We also disprove the conjectured formula for the case $m$ $\equiv 0$ $(mod3)$ appearing in \cite{}