On the domination number of Cartesian product of two directed cycles
Summary: 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 \(m,n\geq 2\). In [Inf. Process. Lett. 110, No. 5, 171--173 (2010; Zbl 1197.05111)], \textit{J. Liu} et al. determined the exact values of \(\gamma(C_m\square C_n)\) for \(m=2,3,4,5,6\). In [Discuss. Math., Graph Theory 33, No. 2, 387--394 (2013; Zbl 1293.05271)], \textit{M. 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, \(\lceil (2k+1)n/2\rceil\leq\gamma(C_{3k+1}\square C_n)\leq \lfloor(2k+1)n/2 \rfloor+k\). 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 \(\gamma(C_{13}\square C_n)\) is provided.
scientific article; zbMATH DE number 6950782