Abstract
For a positive integer k, a k-rainbow dominating function (kRDF) of a digraph D is a function f from the vertex set V(D) to the set of all subsets of \(\{1,2,\ldots ,k\}\) such that for any vertex v with \(f(v)=\emptyset \), \(\bigcup _{u\in N^-_D(v)}f(u)=\{1,2,\ldots ,k\}\), where \(N^-_D(v)\) is the set of in-neighbors of v. The weight of a kRDF f is defined as \(\sum _{v\in V(D)}|f(v)|\). The k-rainbow domination number of a digraph D, denoted by \(\gamma _{rk}(D)\), is the minimum weight of a kRDF of D. We establish several bounds on \(\gamma _{rk}(D)\). Moreover, we determine the exact value of \(\gamma _{rk}(D)\) in Cartesian product of directed cycles for \(k\ge 4\).
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Acknowledgments
This work was supported by NSFC (No. 11471273), the Research Foundation of Education Bureau of Jiangxi Province of China (No. GJJ150561) and the Doctor Fund of East China University of Technology (No. DHBK2015319). The authors would like to thank the anonymous reviewer(s) and editor(s) for their valuable comments and suggestions.
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Hao, G., Qian, J. On the Rainbow Domination Number of Digraphs. Graphs and Combinatorics 32, 1903–1913 (2016). https://doi.org/10.1007/s00373-016-1692-0
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DOI: https://doi.org/10.1007/s00373-016-1692-0