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

Mathematics

Hong Gao¹  Changqing Xi¹  Yuansheng Yang² 

[1] College of Science, Dalian Maritime University, Dalian 116026, China;School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China;

关键词: rainbow domination;    graph domination;    cartesian product graph;    cycle;   

DOI  :  10.3390/math8010065

来源: DOAJ

【 摘 要 】

We have studied the k-rainbow domination number of $C_n\square C_m$ for $k\ge 4$ (Gao et al. 2019), in which we present the 3-rainbow domination number of $C_n\square C_m$ , which should be bounded above by the four-rainbow domination number of $C_n\square C_m$ . Therefore, we give a rough bound on the 3-rainbow domination number of $C_n\square C_m$ . In this paper, we focus on the 3-rainbow domination number of the Cartesian product of cycles, $C_n\square C_m$ . 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 $\omega \left(f\right)={\sum }_{v\in V\left(G\right)}\left|f\left(v\right)\right|$ . The 3-rainbow domination number, ${\gamma }_{r3}\left(G\right)$ , 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 $C_3\square C_m$ and $C_4\square C_m$ and present a tighter bound on the 3-rainbow domination number of $C_n\square C_m$ for $n\ge 5$ .

【 授权许可】

Unknown