On the Domination of Cartesian Product of Directed Cycles: Results for Certain Equivalence Classes of Lengths

Let \( \gamma ( \overrightarrow{C_m} \square \overrightarrow{C_n} ) \) be the domination number of the Cartesian product of directed cycles \( \overrightarrow{C_m} \) and \( \overrightarrow{C_n} \) for $m, n \ge 2 $. Shaheen [13] and Liu et al. ([11], [12]) determined the value of \( \gamma ( \overrightarrow{C_m} \square \overrightarrow{C_n} ) \) when $ m \le 6 $ and [12] when both $m$ and $ n \equiv 0 (\mod 3) $. In this article we give, in general, the value of \( \gamma ( \overrightarrow{C_m} \square \overrightarrow{C_n} ) \) when $ m \equiv 2(\mod 3) $ and improve the known lower bounds for most of the remaining cases. We also disprove the conjectured formula for the case $ m \equiv 0 ( \mod 3) $ appearing in [12].