Skip to main content
SpringerLink
Log in

Relationships between the clique number, chromatic number, and the degree for some graphs

  • Published:
Automatic Control and Computer Sciences Aims and scope Submit manuscript

Abstract

The existence of an independent transversal for the maximal cliques of a graph of a small degree is proved. Some relationships between the clique number, the chromatic number, and the degree for graphs with an n-clique cutset are also deduced.

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. Berlov, S.L., Helly’s Property for n-Cliques and the Degree of a Graph, Zapiski Nauchnykh Seminarov POMI, 2006, vol. 340, pp. 5–9.

    MATH  Google Scholar 

  2. Alon, N., The Strong Chromatic Number of a Graph, Random Struct. Alg, 1992, vol. 3, pp. 1–7.

    Article  MATH  Google Scholar 

  3. Haxell, P.E., On the Strong Chromatic Number, Combinatorics, Probability and Computing, 2004, vol. 13,issue 6, pp. 857–865.

    Article  MathSciNet  MATH  Google Scholar 

  4. Axenovich, M., On the Strong Chromatic Number of Graphs, SIAM Journal on Discrete Mathematics archive, 2006, vol. 20,issue 3, pp. 741–747.

    Article  MathSciNet  MATH  Google Scholar 

  5. Erdös, P., and Gallai, T., On Minimal Number of Vertices Representing the Edges of a Graph, Publ. Math. Inst. Hung. Acad. Sci, 1961, vol. 6, pp. 89–96.

    Google Scholar 

  6. Brooks, R.L. On Coloring the Nodes of a Network, Proc. Cambrige Phil., 1941, vol. 37, pp. 194–197.

    Article  MathSciNet  Google Scholar 

  7. Reed, B., A Strengthening of Brooks’ Theorem, Journal of Combinatorial Theory, Series B, 1999, vol. 76,issue 2, pp. 136–149.

    Article  MathSciNet  MATH  Google Scholar 

  8. Kostochka, A.V., Degree, Density and Chromatic Number of Graphs, Metody Diskret. Analiz., 1980, no. 35, pp. 45–70, 104-105.

  9. Jensen, T., and Toft, B., Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, 1995.

  10. Catlin, P. A., Survey of Extensions of Brooks’ Graph Coloring Theorem, Annals of the New York Academy of Sciences, 1979, vol. 328, p. 95.

    Article  MathSciNet  Google Scholar 

  11. Catlin, P., Brooks’ Graph-Coloring Theorem and the Independence Number, Journal of Combinatorial Theory, Series B, 1979, vol. 27, pp. 42–48.

    Article  MathSciNet  MATH  Google Scholar 

  12. Beutelspacher, A., and Hering, P., Minimal Graphs for Which the Chromatic Number Equals the Maximal Degree, Ara Combinatorica, 1984, vol. 18, pp. 201–216.

    MathSciNet  MATH  Google Scholar 

  13. Borodin, O., and Kostochka, A., On an Upper Bound on a Graph’s Chromatic Number, Depending on the Graphs’s Degree and Density, Journal of Combinatorial Theory, Series B, 1977, vol.23, pp. 247–250.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Physical-Mathematical Lyceum no. 239, Moscow, Russia

    S. L. Berlov

Authors
  1. S. L. Berlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. L. Berlov.

Additional information

Original Russian Text © S.L. Berlov, 2008, published in Modelirovanie i Analiz Informatsionnykh Sistem, 2008, No. 4, pp. 10–22.

About this article

Cite this article

Berlov, S.L. Relationships between the clique number, chromatic number, and the degree for some graphs. Aut. Conrol Comp. Sci. 44, 407–414 (2010). https://doi.org/10.3103/S0146411610070060

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0146411610070060

Keywords

Search

Navigation