The Italian domination numbers of some products of directed cycles

An Italian dominating function on a digraph $D$ with vertex set $V(D)$ is defined as a function $f:V(D)\to \{0,1,2\}$ such that every vertex $v\in V(D)$ with $f(v)=0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w)=2$. In this paper, we determine the exact values of the Italian domination numbers of some products of directed cycles.