On the Domination Number of Cartesian Product of Two Directed Cycles

Abstract

Denote by $\gamma (G)$ the domination number of a digraph $G$ and $C_m\square C_n$ the Cartesian product of $C_m$ and $C_n$, the directed cycles of length $m,n\ge 2$. In 2010, Liu et al. determined the exact values of $\gamma (C_m\square C_n)$ for $m=\mathrm{2,3},\mathrm{4,5},6$. In 2013, Mollard determined the exact values of $\gamma (C_m\square C_n)$ for $m=3k+2$. In this paper, we give lower and upper bounds of $\gamma (C_m\square C_n)$ with $m=3k+1$ for different cases. In particular, $\lceil \left(2k+1\right)n/2\rceil \le \gamma (C_{3k+1}\square C_n)\le \lfloor \left(2k+1\right)n/2\rfloor +k$. Based on the established result, the exact values of $\gamma (C_m\square C_n)$ are determined for $m=7$ and 10 by the combination of the dynamic algorithm, and an upper bound for $\gamma (C_{13}\square Cn)$ is provided.