2-Rainbow domination number of Cartesian products: $$C_{n}\square C_{3}$$ and $$C_{n}\square C_{5}$$

Zofia Stępień and Maciej Zwierzchowski ()
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Zofia Stępień: West Pomeranian University of Technology
Maciej Zwierzchowski: West Pomeranian University of Technology

Journal of Combinatorial Optimization, 2014, vol. 28, issue 4, No 3, 748-755

Abstract: Abstract 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 $$ \bigcup \nolimits _{u\in N(v)}f(u)=\{1,\ldots ,k\},$$ 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 $$\gamma _{r2}(C_{n}\square C_{5})=2n.$$

Keywords: Domination; Rainbow domination; Cartesian product of graphs (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10878-012-9582-8

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