Aequationes mathematicae

, Volume 82, Issue 1–2, pp 25–34 | Cite as

The {k}-domatic number of a graph

  • D. Meierling
  • S. M. Sheikholeslami
  • L. Volkmann


For a positive integer k, a {k}-dominating function of a graph G is a function f from the vertex set V(G) to the set {0, 1, 2, . . . , k} such that for any vertex \({v\in V(G)}\), the condition \({\sum_{u\in N[v]}f(u)\ge k}\) is fulfilled, where N[v] is the closed neighborhood of v. A {1}-dominating function is the same as ordinary domination. A set {f 1, f 2, . . . , f d } of {k}-dominating functions on G with the property that \({\sum_{i=1}^df_i(v)\le k}\) for each \({v\in V(G)}\), is called a {k}-dominating family (of functions) on G. The maximum number of functions in a {k}-dominating family on G is the {k}-domatic number of G, denoted by d {k}(G). Note that d {1}(G) is the classical domatic number d(G). In this paper we initiate the study of the {k}-domatic number in graphs and we present some bounds for d {k}(G). Many of the known bounds of d(G) are immediate consequences of our results.

Mathematics Subject Classification (2000)



{k}-dominating function {k}-domination number {k}-domatic number 


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • D. Meierling
    • 1
  • S. M. Sheikholeslami
    • 2
  • L. Volkmann
    • 1
  1. 1.Lehrstuhl II für MathematikRWTH Aachen UniversityAachenGermany
  2. 2.Department of MathematicsAzarbaijan University of Tarbiat MoallemTabrizIslamic Republic of Iran

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