2-Rainbow domination number of Cartesian products: $$C_{n}\square C_{3}$$ and $$C_{n}\square C_{5}$$
Zofia Stępień, Maciej Zwierzchowski · Journal of Combinatorial Optimization · 2012
A function $$f:V(G)\rightarrow \mathcal P (\{1,\ldots ,k\})$$ is called a $$k$$ -rainbow dominating function of $$G$$ (for short $$kRDF$$ of $$G)$$ if $$ \bigcup olimits _{u\in N(v)}f(u)=\{1,\ldots ,k\},$$ for each vertex $$ v\in V(G)$$ with $$f(v)=\varnothing .$$ By $$w(f)$$ we mean $$\sum _{v\in V(G)}\left|f(v)\right|$$ and we call it the weight of $$f$$ in $$G.$$ The minimum weight of a $$ kRDF$$ of $$G$$ is called the $$k$$ -rainbow domination number of $$G$$ and it is denoted by $$\gamma _{rk}(G).$$ We investigate the $$2$$ -rainbow domination number of Cartesian products of cycles. We give the exact value of the $$2$$ -rainbow domination number of $$C_{n}\square C_{3}$$ and we give the estimation of this number with respect to $$C_{n}\square C_{5},$$ $$(n\ge 3).$$ Additionally, for $$n=3,4,5,6,$$ we show that $$\gamma _{r2}(C_{n}\square C_{5})=2n.$$