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The k-Rainbow Domination and Domatic Numbers of Digraphs

Sheikholeslami, S.M.;Volkmann, Lutz

  • Received : 2013.11.17
  • Accepted : 2014.07.14
  • Published : 2016.03.23

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 $d_{rk}(D)$.

Keywords

Digraph;k-rainbow dominating function;k-rainbow domination number;k-rainbow domatic number

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Cited by

  1. Rainbow reinforcement numbers in digraphs vol.10, pp.01, 2017, https://doi.org/10.1142/S1793557117500048

Acknowledgement

Supported by : Azarbaijan Shahid Madani University