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Bounds on the 2-Rainbow Domination Number of Graphs

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Abstract

A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any \({v\in V(G), f(v)=\emptyset}\) implies \({\bigcup_{u\in N(v)}f(u)=\{1,2\}.}\) The 2-rainbow domination number γ r2(G) of a graph G is the minimum \({w(f)=\Sigma_{v\in V}|f(v)|}\) over all such functions f. Let G be a connected graph of order |V(G)| = n ≥ 3. We prove that γ r2(G) ≤ 3n/4 and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of γ r2(G) in terms of diameter are also given.

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Authors and Affiliations

  1. Department of Mathematics, Southeast University, Nanjing, 211189, China

    Yunjian Wu

  2. Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

    Nader Jafari Rad

Authors
  1. Yunjian Wu
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  2. Nader Jafari Rad
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Correspondence to Yunjian Wu.

Additional information

The work is supported by NSFC under Grant No. 11126054.

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Wu, Y., Rad, N.J. Bounds on the 2-Rainbow Domination Number of Graphs. Graphs and Combinatorics 29, 1125–1133 (2013). https://doi.org/10.1007/s00373-012-1158-y

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  • DOI: https://doi.org/10.1007/s00373-012-1158-y

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