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The 2-Rainbow Domination Numbers of \({\boldsymbol{C}}_4\Box {\boldsymbol{C}}_n\) and \({\boldsymbol{C}}_8\Box {\boldsymbol{C}}_n\)

  • Zehui Shao
  • Zepeng LiEmail author
  • Rija Erveš
  • Janez Žerovnik
Short Communication
  • 3 Downloads

Abstract

A k-rainbow dominating function (kRDF) of G is a function \(f:V(G)\rightarrow {\mathcal {P}}(\{1,2,\ldots ,k\})\) for which \(f(v)=\emptyset \) we have \(\bigcup \nolimits _{u\in N(v)}f(u)=\{1,2,\ldots ,k\}\). The weight w(f) of a function f is defined as \(w(f)=\sum _{v\in V(G)}\left| f(v)\right| \). The minimum weight of a kRDF of G is called the k-rainbow domination number of G, which is denoted by \(\gamma _{rk}(G)\). In this paper, we determine the exact values of the 2-rainbow domination numbers of \(C_4\Box C_n\) and \(C_8\Box C_n\). It follows that \(\gamma _{r2}\not = 2\gamma \) for graphs \(C_4\Box C_n\) (\(n \ge 4\)) and \(C_8\Box C_n\) (\(n \ge 8\)), answering in part a question raised by Brešar.

Keywords

2-rainbow domination Domination number Cartesian product Cycle 

Notes

Acknowledgements

The authors thank anonymous referees sincerely for their careful review and helpful suggestions to improve this manuscript. The work of Z. Li was partially supported by National Natural Science Foundation of China under Grants 61802158, 61672050 and the Fundamental Research Funds for the Central Universities under grant lzujbky-2018-37. The work of Z. Shao was partially supported by the National Key Research and Development Program under grant 2017YFB0802300 and the National Science Foundation of Guangdong Province under Grant 2018A0303130115. Two authors (R.E. and J.Ž) were supported partially by ARRS, the Research Agency of Slovenia.

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Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  • Zehui Shao
    • 1
  • Zepeng Li
    • 2
    Email author
  • Rija Erveš
    • 3
  • Janez Žerovnik
    • 4
    • 5
  1. 1.Institute of Computing Science and TechnologyGuangzhou UniversityGuangzhouChina
  2. 2.School of Information Science and EngineeringLanzhou UniversityLanzhouChina
  3. 3.FCETEAUniversity of MariborMariborSlovenia
  4. 4.FMEUniversity of LjubljanaLjubljanaSlovenia
  5. 5.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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