Skip to main content
SpringerLink
Log in

The k-domatic number of a graph

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-dominating set of the graph G is a subset D of V (G) such that every vertex of V (G)-D is adjacent to at least k vertices in D. A k-domatic partition of G is a partition of V (G) into k-dominating sets. The maximum number of dominating sets in a k-domatic partition of G is called the k-domatic number d k (G).

In this paper, we present upper and lower bounds for the k-domatic number, and we establish Nordhaus-Gaddum-type results. Some of our results extend those for the classical domatic number d(G) = d 1(G).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. J. Cockayne and S. T. Hedetniemi: Towards a theory of domination in graphs. Networks 7 (1977), 247–261.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. F. Fink and M. S. Jacobson: n-domination in graphs. Graph Theory with Applications to Algorithms and Computer Science. John Wiley and Sons, New York (1985), 282–300.

    Google Scholar 

  3. J. F. Fink and M. S. Jacobson: On n-domination, n-dependence and forbidden subgraphs. Graph Theory with Applications to Algorithms and Computer Science. John Wiley and Sons, New York (1985), 301–311.

    Google Scholar 

  4. T. W. Haynes, S. T. Hedetniemi and P. J. Slater: Fundamentals of Domination in Graphs. Marcel Dekker, New York, 1998, pp. 233–269.

    MATH  Google Scholar 

  5. T. W. Haynes, S. T. Hedetniemi and P. J. Slater (eds.): Domination in Graphs: Advanced Topics. Marcel Dekker, New York, 1998.

    MATH  Google Scholar 

  6. B. Zelinka: Domatic number and degrees of vertices of a graph. Math. Slovaca 33 (1983), 145–147.

    MATH  MathSciNet  Google Scholar 

  7. B. Zelinka: Domatic numbers of graphs and their variants: A survey Domination in Graphs: Advanced Topics. Marcel Dekker, New York, 1998, pp. 351–374.

    Google Scholar 

  8. B. Zelinka: On k-ply domatic numbers of graphs. Math. Slovaca 34 (1984), 313–318.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Lehrstuhl II für Mathematik, RWTH Aachen University, 52056, Aachen, Germany

    Karsten Kämmerling & Lutz Volkmann

Authors
  1. Karsten Kämmerling
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lutz Volkmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lutz Volkmann.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kämmerling, K., Volkmann, L. The k-domatic number of a graph. Czech Math J 59, 539–550 (2009). https://doi.org/10.1007/s10587-009-0036-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-009-0036-0

Keywords

MSC 2000

Search

Navigation