Domination number of Cartesian products of directed cycles

Let γ(G)γ(G) denote the domination number of a digraph G   and let Cm□CnCm□Cn denote the Cartesian product of CmCm and CnCn, the directed cycles of length m,n⩾2m,n⩾2. In Liu et al. (2010) [11], we determined the exact values of γ(Cm□Cn)γ(Cm□Cn) when m=2,3,4m=2,3,4. In this paper, we give a lower and upper bounds for γ(Cm□Cn)γ(Cm□Cn). Furthermore, we prove a necessary and sufficient conditions for Cm□CnCm□Cn to have an efficient dominating set. Also, we determine the exact values: γ(C5□Cn)=2nγ(C5□Cn)=2n; γ(C6□Cn)=2nγ(C6□Cn)=2n if n≡0n≡0(mod 3)(mod 3), otherwise, γ(C6□Cn)=2n+2γ(C6□Cn)=2n+2; γ(Cm□Cn)=mn3 if m≡0m≡0(mod 3)(mod 3) and n≡0n≡0(mod 3)(mod 3).

Research highlights
► A necessary and sufficient conditions for Cm□CnCm□Cn to have an efficient dominating set has been proved.
► Determine the exact values of γ(C5□Cn)γ(C5□Cn) and γ(C6□Cn)γ(C6□Cn).
► A lower and upper bounds for γ(Cm□Cn)γ(Cm□Cn) is given.