On Total Domination Number of Cartesian Product of Directed Cycles

Abstract

Let ${\gamma }_t(D)$ denote the total domination number of a digraph $D$, and let $C_m\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n$ denote the Cartesian product graph of $C_m$ and $C_n$, where $C_m$ denotes the directed cycle of length $m$, $m\le n$. In [On domination number of Cartesian product of directed cycles, Information Processing Letters 110 (2010) 171-173], Liu et al. determined the domination number of $C_2\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n$, $C_3\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n$, and $C_4\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n$. In this paper, we determine the exact values of ${\gamma }_t(C_m\mathrm{<img\; draggable="false"\; role="img"\; class="emoji"\; alt="◻"\; src="https://s.w.org/images/core/emoji/15.0.3/svg/25fb.svg">}{C}_n)$ when at least one of $m$ and $n$ is even, or $n$ is odd and $m=1,3,5,$ or $7$.