On $ {L}(2,1) $-labelings of some products of oriented cycles

We refine two results of Jiang, Shao and Vesel on the \begin{document}$ L(2,1) $\end{document} -labeling number \begin{document}$ \lambda $\end{document} of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of \begin{document}$ \lambda(\overrightarrow{C_m} \square \overrightarrow{C_n}) $\end{document} for \begin{document}$ m $\end{document} , \begin{document}$ n \geq 40 $\end{document} ; in the case of strong product, we either compute the exact value or establish a gap of size one for \begin{document}$ \lambda(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n}) $\end{document} for \begin{document}$ m $\end{document} , \begin{document}$ n \geq 48 $\end{document} .