What must and what need not be contained in a graph of uncountable chromatic number?

Abstract

We investigate the following problem: What countable graphs must a graph of uncountable chromatic number contain? We define two graphsΓ andΔ which are very similar and we show thatΓ is contained in every graph of uncountable chromatic number, whileΔ is (consistently) not.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    P. Erdős, Graph theory and probability,Canad. J. Math. 11 (1959), 34–38.

    MathSciNet  Google Scholar 

  2. [2]

    P. Erdős, F. Galvin andA. Hajnal, On set systems having large chromatic numbers and not containing prescribed subsystems,Coll. Math. Soc. János Bolyai 10,Infinite and Finite Sets, Keszthely (Hungary) 1973, 425–513.

    Google Scholar 

  3. [3]

    P. Erdős andA. Hajnal, On chromatic number of graphs and set-systems,Acta Math. Acad. Sci. Hung. 17 (1966), 61–99.

    Article  Google Scholar 

  4. [4]

    P. Erdős, A. Hajnal andS. Shelah, On some general properties of chromatic numbers,Coll. Math. Soc. János Bolyai 8,Topics in Topology, Keszthely, (Hungary) 1973, 243–255.

    Google Scholar 

  5. [5]

    P. Erdős andR. Rado, A construction of graphs without triangles having pre-assigned order and chromatic number,Journal of the London Math. Soc. 35 (1960), 445–448.

    Article  Google Scholar 

  6. [6]

    A. Hajnal andA. Máté, Set mappings, partitions and chromatic numbers,Proceedings of Bristol Logic Conference, July 1973, 67–69.

  7. [7]

    P. Komjáth, A note on Hajnal—Máté graphs,Studia Sci. Math. Hung. 15 (1981), 275–278.

    Google Scholar 

  8. [8]

    L. Lovász, On chromatic number of graphs and set-systems,Acta Math. Acad. Sci. Hung. 19 (1969), 59–67.

    Article  Google Scholar 

  9. [9]

    J. Mycielski, Sur le coloriage des graphs,Colloq. Math. 3 (1955), 161–162.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    C. Thomassen, Cycles in graphs of uncountable chromatic number,Combinatorica 3 (1983), 133–134.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    A. A. Zykov, On some properties of linear complexes,Russian Math. Sbornik, N. S. 24 (66) (1949), 163–188.

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Dedicated to Paul Erdős on his seventieth birthday

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hajnal, A., Komjáth, P. What must and what need not be contained in a graph of uncountable chromatic number?. Combinatorica 4, 47–52 (1984). https://doi.org/10.1007/BF02579156

Download citation

AMS subject classification (1980)

  • 04 A 20
  • 03 E 05
  • 03 E 50
  • 04 A 30