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

Abstract

<p style='text-indent:20px;'>We refine two results of Jiang, Shao and Vesel on the <inline-formula><tex-math id="M2">\begin{document}$ L(2,1) $\end{document}</tex-math></inline-formula>-labeling number <inline-formula><tex-math id="M3">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula> of the Cartesian and the strong product of two oriented cycles. For the Cartesian product, we compute the exact value of <inline-formula><tex-math id="M4">\begin{document}$ \lambda(\overrightarrow{C_m} \square \overrightarrow{C_n}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ n \geq 40 $\end{document}</tex-math></inline-formula>; in the case of strong product, we either compute the exact value or establish a gap of size one for <inline-formula><tex-math id="M7">\begin{document}$ \lambda(\overrightarrow{C_m} \boxtimes \overrightarrow{C_n}) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M8">\begin{document}$ m $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ n \geq 48 $\end{document}</tex-math></inline-formula>.</p>