DMGT

Discussiones Mathematicae Graph Theory 32(1) (2012) 129-140
DOI: https://doi.org/10.7151/dmgt.1591

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, …,f_d} of k-rainbow dominating functions on G with the property that ∑_{i = 1}^d |f_i(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 d_rk(G). Note that d_r1(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 d_rk(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.

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