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The 3-Rainbow Domination Number of the Cartesian Product of Cycles

Author

Listed:
  • Hong Gao

    (College of Science, Dalian Maritime University, Dalian 116026, China)

  • Changqing Xi

    (College of Science, Dalian Maritime University, Dalian 116026, China)

  • Yuansheng Yang

    (School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China)

Abstract

We have studied the k -rainbow domination number of C n □ C m for k ≥ 4 (Gao et al. 2019), in which we present the 3-rainbow domination number of C n □ C m , which should be bounded above by the four-rainbow domination number of C n □ C m . Therefore, we give a rough bound on the 3-rainbow domination number of C n □ C m . In this paper, we focus on the 3-rainbow domination number of the Cartesian product of cycles, C n □ C m . A 3-rainbow dominating function (3RDF) f on a given graph G is a mapping from the vertex set to the power set of three colors { 1 , 2 , 3 } in such a way that every vertex that is assigned to the empty set has all three colors in its neighborhood. The weight of a 3RDF on G is the value ω ( f ) = ∑ v ∈ V ( G ) | f ( v ) | . The 3-rainbow domination number, γ r 3 ( G ) , is the minimum weight among all weights of 3RDFs on G . In this paper, we determine exact values of the 3-rainbow domination number of C 3 □ C m and C 4 □ C m and present a tighter bound on the 3-rainbow domination number of C n □ C m for n ≥ 5 .

Suggested Citation

  • Hong Gao & Changqing Xi & Yuansheng Yang, 2020. "The 3-Rainbow Domination Number of the Cartesian Product of Cycles," Mathematics, MDPI, vol. 8(1), pages 1-20, January.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:1:p:65-:d:304487
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    References listed on IDEAS

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    1. Brezovnik, Simon & Šumenjak, Tadeja Kraner, 2019. "Complexity of k-rainbow independent domination and some results on the lexicographic product of graphs," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 214-220.
    2. Zepeng Li & Zehui Shao & Jin Xu, 2018. "Weak {2}-domination number of Cartesian products of cycles," Journal of Combinatorial Optimization, Springer, vol. 35(1), pages 75-85, January.
    3. Zepeng Li & Zehui Shao & Pu Wu & Taiyin Zhao, 2019. "On the 2-rainbow domination stable graphs," Journal of Combinatorial Optimization, Springer, vol. 37(4), pages 1327-1341, May.
    4. Zofia Stȩpień & Lucjan Szymaszkiewicz & Maciej Zwierzchowski, 2015. "The Cartesian product of cycles with small 2-rainbow domination number," Journal of Combinatorial Optimization, Springer, vol. 30(3), pages 668-674, October.
    5. Michel Mollard, 2014. "The domination number of Cartesian product of two directed paths," Journal of Combinatorial Optimization, Springer, vol. 27(1), pages 144-151, January.
    Full references (including those not matched with items on IDEAS)

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