Oriented Chromatic Number of Cartesian Products $ P_m \square P_n $ and $ C_m \square P_n $

We consider oriented chromatic number of Cartesian products of two paths $ P_m \square P_n $ and of Cartesian products of paths and cycles, $ C_m \square P_n $. We say that the oriented graph \( \overrightarrow{G} \) is colored by an oriented graph \( \overrightarrow{H} \) if there is a homomorphism from \( \overrightarrow{G} \) to \( \overrightarrow{H} \). In this paper we show that there exists an oriented tournament \( \overrightarrow{H}_{10} \) with ten vertices which colors every orientation of $ P_8 \square P_n $ and every orientation of $ C_m \square P_n $, for $m = 3, 4, 5, 6, 7$ and $n \ge 1 $. We also show that there exists an oriented graph \( \overrightarrow{T}_{16} \) with sixteen vertices which colors every orientation of $ C_m \square P_n $.