Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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The k-Rainbow Domination and Domatic Numbers of Digraphs

  • Sheikholeslami, S.M. (Department of Mathematics, Azarbaijan Shahid Madani University) ;
  • Volkmann, Lutz (Lehrstuhl II fur Mathematik, RWTH Aachen University)
  • Received : 2013.11.17
  • Accepted : 2014.07.14
  • Published : 2016.03.23
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Abstract

For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set $\{1,2,{\ldots},k\}$ such that for any vertex $v{\in}V(D)$ with $f(v)={\emptyset}$ the condition ${\cup}_{u{\in}N^-(v)}$ $f(u)=\{1,2,{\ldots},k\}$ is fulfilled, where $N^-(v)$ is the set of in-neighbors of v. A set $\{f_1,f_2,{\ldots},f_d\}$ of k-rainbow dominating functions on D with the property that $\sum_{i=1}^{d}{\mid}f_i(v){\mid}{\leq}k$ for each $v{\in}V(D)$, is called a k-rainbow dominating family (of functions) on D. The maximum number of functions in a k-rainbow dominating family on D is the k-rainbow domatic number of D, denoted by $d_{rk}(D)$. In this paper we initiate the study of the k-rainbow domatic number in digraphs, and we present some bounds for $d_{rk}(D)$.

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References

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