笛卡尔乘积有向图C_2×C_n与C_3×C_n的3-彩虹控制数

The 3-rainbow domination number of the Cartesian products digraph C_2×C_n and C_3×C_n

设γ_(rk)(D)是有向图D的k-彩虹控制数且设C_m×C_n是m长有向圈C_m与n长有向圈C_n的笛卡尔乘积有向图.用构造的方法找到了笛卡尔乘积有向图C_2×C_n与C_3×C_n的3-彩虹控制数的上界,并证明了此上界恰好为其下界,即得到了γ_(r3)(C_2×C_n)与γ_(r3)(C_3×C_n)的精确值.

Let γ（rk）（D）be the k-rainbow domination number of a digraph D,and let Cm ×Cn be the Cartesian product digraph of the directed cycle Cm of length mand the directed cycle Cn of length n.The upper bounds on 3-rainbow domination number of C2×Cn and C3×Cn are found by construction methods,which are exactly the lower bounds,that is,the exact values of γ（r3）（C2×Cn）and γ（r3）（C3×Cn）are obtained.