On the Domination Number of Cartesian Product of Two Directed Cycles

[摘要] Denote byγ(G)the domination number of a digraphGandCm□Cnthe Cartesian product ofCmandCn, the directed cycles of lengthm,n≥2. In 2010, Liu et al. determined the exact values ofγ(Cm□Cn)form=2,3,4,5,6. In 2013, Mollard determined the exact values ofγ(Cm□Cn)form=3k+2. In this paper, we give lower and upper bounds ofγ(Cm□Cn)withm=3k+1for different cases. In particular,⌈2k+1n/2⌉≤γ(C3k+1□Cn)≤⌊2k+1n/2⌋+k. Based on the established result, the exact values ofγ(Cm□Cn)are determined form=7and 10 by the combination of the dynamic algorithm, and an upper bound forγ(C13□Cn)is provided.

[学科分类] 应用数学