On the (total) Roman domination in Latin square graphs

Latin square, also known as Latin square matrix, refers to a kind of $n\times n$ matrix, in which there are exactly $n$ different symbols and each symbol appears exactly once in each row and column. A Latin square graph $\mathrm{\Gamma }(L)$ is a simple graph associated with a Latin square $L$. This paper studied the relationships between the (total) Roman domination number and (total) domination number of Latin square graph $\mathrm{\Gamma }(L)$. We showed that ${\gamma }_R(\mathrm{\Gamma }(L))=2\gamma (\mathrm{\Gamma }(L))$ or ${\gamma }_R(\mathrm{\Gamma }(L))=2\gamma (\mathrm{\Gamma }(L))-1$, and ${\gamma }_{tR}(\mathrm{\Gamma }(L))\ge \frac{8{\gamma }_t(\mathrm{\Gamma }(L))}{5}$ for $n\ge 2$. In 2021, Pahlavsay et al. proved $\gamma (\mathrm{\Gamma }(L))\ge \lceil \frac{n}{2}\rceil$ and ${\gamma }_t(\mathrm{\Gamma }(L))\ge \lceil \frac{4n-2}{7}\rceil$ for $n\ge 2$. In this paper, we showed that ${\gamma }_R(\mathrm{\Gamma }(L))\ge 2\lceil \frac{n}{2}\rceil$ (equality holds if, and only if, $\gamma (\mathrm{\Gamma }(L))=\lceil \frac{n}{2}\rceil$) and ${\gamma }_t(\mathrm{\Gamma }(L))>\frac{4n}{7}$ for $n\ge 2$. Since ${\gamma }_R(G)\le 2\gamma (G)$ and ${\gamma }_{tR}(G)\le 2{\gamma }_t(G)$ for any graph $G$, our results can deduce or improve Pahlavsay et al.'s results. Moreover, we characterized these Latin squares for ${\gamma }_R(\mathrm{\Gamma }(L))=2\lceil \frac{n}{2}\rceil$, which is equal to $\gamma (\mathrm{\Gamma }(L))=\lceil \frac{n}{2}\rceil$.