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

by 1,*, 1 and
1
College of Science, Dalian Maritime University, Dalian 116026, China
2
School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 65; https://doi.org/10.3390/math8010065
Received: 11 December 2019 / Revised: 26 December 2019 / Accepted: 30 December 2019 / Published: 2 January 2020
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 $n ≥ 5$ . View Full-Text
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Gao, H.; Xi, C.; Yang, Y. The 3-Rainbow Domination Number of the Cartesian Product of Cycles. Mathematics 2020, 8, 65.