The k-Rainbow Domination and Domatic Numbers of Digraphs

• Journal title : Kyungpook mathematical journal
• Volume 56, Issue 1,  2016, pp.69-81
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2016.56.1.69
Title & Authors
The k-Rainbow Domination and Domatic Numbers of Digraphs
Sheikholeslami, S.M.; Volkmann, Lutz;

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 $\small{\{1,2,{\ldots},k\}}$ such that for any vertex $\small{v{\in}V(D)}$ with $\small{f(v)={\emptyset}}$ the condition $\small{{\cup}_{u{\in}N^-(v)}}$ $\small{f(u)=\{1,2,{\ldots},k\}}$ is fulfilled, where $\small{N^-(v)}$ is the set of in-neighbors of v. A set $\small{\{f_1,f_2,{\ldots},f_d\}}$ of k-rainbow dominating functions on D with the property that $\small{\sum_{i=1}^{d}{\mid}f_i(v){\mid}{\leq}k}$ for each $\small{v{\in}V(D)}$, 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 $\small{d_{rk}(D)}$. In this paper we initiate the study of the k-rainbow domatic number in digraphs, and we present some bounds for $\small{d_{rk}(D)}$.
Keywords
Digraph;k-rainbow dominating function;k-rainbow domination number;k-rainbow domatic number;
Language
English
Cited by
1.
Rainbow reinforcement numbers in digraphs, Asian-European Journal of Mathematics, 2016, 1750004
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