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The k-Rainbow Domination Number of CnCm

by Hong Gao 1,*, Kun Li 1 and Yuansheng Yang 2
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 2019, 7(12), 1153; https://doi.org/10.3390/math7121153
Received: 10 October 2019 / Revised: 20 November 2019 / Accepted: 21 November 2019 / Published: 1 December 2019
(This article belongs to the Section Mathematics and Computer Science)
Given a graph G and a set of k colors, assign an arbitrary subset of these colors to each vertex of G. If each vertex to which the empty set is assigned has all k colors in its neighborhood, then the assignment is called a k-rainbow dominating function (kRDF) of G. The minimum sum of numbers of assigned colors over all vertices of G is called the k-rainbow domination number of graph G, denoted by γ r k ( G ) . In this paper, we focus on the study of the k-rainbow domination number of the Cartesian product of cycles, C n C m . For k 8 , based on the results of J. Amjadi et al. (2017), γ r k ( C n C m ) = m n . For ( 4 k 7 ) , we give a proof for the new lower bound of γ r 4 ( C n C 3 ) . We construct some novel and recursive kRDFs which are good enough and upon these functions we get sharp upper bounds of γ r k ( C n C m ) . Therefore, we obtain the following results: (1) γ r 4 ( C n C 3 ) = 2 n ; (2) γ r k ( C n C m ) = k m n 8 for n 0 ( mod 4 ) , m 0 ( mod 4 ) ( 4 k 7 ) ; (3) for n 0 ( mod 4 ) or m 0 ( mod 4 ) , m n 2 γ r 4 ( C n C m ) m n 2 + m + n 2 1 and k m n 8 γ r k ( C n C m ) k m n + n 8 + m for 5 k 7 . We also discuss Vizing’s conjecture on the k-rainbow domination number of C n C m . View Full-Text
Keywords: rainbow domination; k-rainbow domination; cartesian product graph; cycle rainbow domination; k-rainbow domination; cartesian product graph; cycle
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Gao, H.; Li, K.; Yang, Y. The k-Rainbow Domination Number of CnCm. Mathematics 2019, 7, 1153.

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