The Colouring Number of Infinite Graphs

Abstract

We show that, given an infinite cardinal μ, a graph has colouring number at most μ if and only if it contains neither of two types of subgraph. We also show that every graph with infinite colouring number has a well-ordering of its vertices that simultaneously witnesses its colouring number and its cardinality.

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References

  1. [1]

    B. Dushnik and E. W. Miller: Partially ordered sets, Amer. J. Math.63 (1941), 600–610.

    MathSciNet  Article  Google Scholar 

  2. [2]

    P. Erdős and A. Hajnal, A.: On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hungar17 (1966), 61–99.

    MathSciNet  Article  Google Scholar 

  3. [3]

    R. Halin. Graphentheorie, Wissenschaftliche Buchgesellschaft, Darmstadt, 2 edition, 1989.

    Google Scholar 

  4. [4]

    P. Komjáth: Infinite graphs, Research Monograph. In Preparation.

  5. [5]

    P. Komjáth: A note on uncountable chordal graphs, Discrete Math.338 (2015), 1565–1566.

    MathSciNet  Article  Google Scholar 

  6. [6]

    P. Komjáth: Hadwiger's conjecture for uncountable graphs, Abh. Math. Semin. Univ. Hambg.87 (2017), 337–341.

    MathSciNet  Article  Google Scholar 

  7. [7]

    K. Kunen: Set theory, volume 34 of Studies in Logic (London), College Publications, London, 2011.

    Google Scholar 

  8. [8]

    S. Shelah: A compactness theorem for singular cardinals, free algebras, whitehead problem and transversals, Israel J. Math.21 (1975), 319–349.

    MathSciNet  Article  Google Scholar 

  9. [9]

    S. Shelah: Notes on partition calculus, Colloq. Math. Soc. János Bolyai, Vol. 10. 1975, 1257–1276.

    MathSciNet  MATH  Google Scholar 

  10. [10]

    W. Sierpiński: Sur un problème de la théorie des relations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2)2 (1933), 285–287.

    MATH  Google Scholar 

Download references

Acknowledgments

We thank the first referee of this paper for pointing out to mention Theorem 1.5 in the Introduction and Lemma 2.7.

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Correspondence to Nathan Bowler or Johannes Carmesin or Péter Komjáth or Christian Reiher.

Additional information

This research was supported by Thematic Excellence Programme, Industry and Digitization Subprogramme, NRDI Office, 2019.

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Bowler, N., Carmesin, J., Komjáth, P. et al. The Colouring Number of Infinite Graphs. Combinatorica 39, 1225–1235 (2019). https://doi.org/10.1007/s00493-019-4045-9

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Mathematics Subject Classification (2010)

  • 05C63
  • 05C75
  • 05C15
  • 03E05