2-rainbow domination of generalized Petersen graphs P(n,2)

  • Authors:

  • Chunling Tong;Xiaohui Lin;Yuansheng Yang;Meiqin Luo

  • Affiliations:

  • Department of Computer Science and Engineering, Dalian University of Technology, Dalian, 116024, PR China and Department of Information Science and Engineering, Shandong Jiaotong University, Jinan ...;Department of Computer Science and Engineering, Dalian University of Technology, Dalian, 116024, PR China;Department of Computer Science and Engineering, Dalian University of Technology, Dalian, 116024, PR China;Department of Computer Science and Engineering, Dalian University of Technology, Dalian, 116024, PR China

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

Assume we have a set of k colors and we assign an arbitrary subset of these colors to each vertex of a graph G. If we require that each vertex to which an empty set is assigned has in its neighborhood all k colors, then this assignment is called the k-rainbow dominating function of a graph G. The corresponding invariant @c"r"k(G), which is the minimum sum of numbers of assigned colors over all vertices of G, is called the k-rainbow domination number of G. Bresar and S@?umenjak [B. Bresar, T.K. S@?umenjak On the 2-rainbow domination in graphs, Discrete Applied Mathematics, 155 (2007) 2394-2400] showed that @?4n5@?@?@c"r"2(P(n,2))@?@?4n5@?+@a, where @a=0 for n=3,9mod10 and @a=1 for n=1,5,7 mod 10. And they raised the question: Is @c"r"2(P(2k+1,k))=2k+1 for all k=2? In this paper, we put forward the answer to the question. More over, we show that @c"r"2(P(n,2))=@?4n5@?+@a, where @a=0 for n=0,3,4,9mod10 and @a=1 for n=1,2,5,6,7,8mod10.