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The k-Rainbow Domination Number of C_{n}□C_{m}

^{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.

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 ${\gamma}_{rk}\left(G\right)$ . In this paper, we focus on the study of the k-rainbow domination number of the Cartesian product of cycles, ${C}_{n}\square {C}_{m}$ . For $k\ge 8$ , based on the results of J. Amjadi et al. (2017), ${\gamma}_{rk}({C}_{n}\square {C}_{m})=mn$ . For $(4\le k\le 7)$ , we give a proof for the new lower bound of ${\gamma}_{r4}({C}_{n}\square {C}_{3})$ . We construct some novel and recursive kRDFs which are good enough and upon these functions we get sharp upper bounds of ${\gamma}_{rk}({C}_{n}\square {C}_{m})$ . Therefore, we obtain the following results: (1) ${\gamma}_{r4}({C}_{n}\square {C}_{3})=2n$ ; (2) ${\gamma}_{rk}({C}_{n}\square {C}_{m})=\frac{kmn}{8}$ for $n\equiv 0(mod4),m\equiv 0(mod4)$ $(4\le k\le 7)$ ; (3) for $n\not\equiv 0(mod4)$ or $m\not\equiv 0(mod4)$ , $\frac{mn}{2}\le {\gamma}_{r4}({C}_{n}\square {C}_{m})\le \frac{mn}{2}+m+\frac{n}{2}-1$ and $\frac{kmn}{8}\le {\gamma}_{rk}({C}_{n}\square {C}_{m})\le k\frac{mn+n}{8}+m$ for $5\le k\le 7$ . We also discuss Vizing’s conjecture on the k-rainbow domination number of ${C}_{n}\square {C}_{m}$ .
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**MDPI and ACS Style**

Gao, H.; Li, K.; Yang, Y. The *k*-Rainbow Domination Number of *C** _{n}*□

*C*

*.*

_{m}*Mathematics*

**2019**,

*7*, 1153.

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