# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## The k-rainbow Domatic Number of a Graph

 Seyyed Mahmoud Sheikholeslami Department of Mathematics Azarbaijan Univercity of Tarbiat Moallem Tarbriz, I.R. Iran Lutz Volkmann Lehrstuhl II für Mathematik RWTH Aachen University 52056 Aachen, Germany

## Abstract

For a positive integer k, a k-rainbow dominating function of a graph G is a function f from the vertex set V(G) to the set of all subsets of the set {1,2, …,k} such that for any vertex v ∈ V(G) with f(v) = ∅ the condition ∪u ∈ N(v)f(u) = {1,2, …,k} is fulfilled, where N(v) is the neighborhood of v. The 1-rainbow domination is the same as the ordinary domination. A set {f1,f2, …,fd} of k-rainbow dominating functions on G with the property that ∑i = 1d |fi(v) | ≤ k for each v ∈ V(G), is called a k-rainbow dominating family (of functions) on G. The maximum number of functions in a k-rainbow dominating family on G is the k-rainbow domatic number of G, denoted by drk(G). Note that dr1(G) is the classical domatic number d(G). In this paper we initiate the study of the k-rainbow domatic number in graphs and we present some bounds for drk(G). Many of the known bounds of d(G) are immediate consequences of our results.

Keywords: k-rainbow dominating function, k-rainbow domination number, k-rainbow domatic number

2010 Mathematics Subject Classification: 05C69.

## References

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Received 10 September 2010
Revised 10 March 2011
Accepted 15 March 2011