Domination number of Cartesian products of directed cycles

Abstract

Let γ ( G ) denote the domination number of a digraph G and let C m □ C n denote the Cartesian product of C m and C n , the directed cycles of length m , n ⩾ 2 . In Liu et al. (2010) [11], we determined the exact values of γ ( C m □ C n ) when m = 2 , 3 , 4 . In this paper, we give a lower and upper bounds for γ ( C m □ C n ) . Furthermore, we prove a necessary and sufficient conditions for C m □ C n to have an efficient dominating set. Also, we determine the exact values: γ ( C 5 □ C n ) = 2 n ; γ ( C 6 □ C n ) = 2 n if n ≡ 0 ( mod 3 ) , otherwise, γ ( C 6 □ C n ) = 2 n + 2 ; γ ( C m □ C n ) = m n 3 if m ≡ 0 ( mod 3 ) and n ≡ 0 ( mod 3 ) .