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The Colouring Number of Infinite Graphs

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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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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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Authors and Affiliations

  1. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, D-20146, Hamburg, Germany

    Nathan Bowler & Christian Reiher

  2. Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, USA

    Johannes Carmesin

  3. Eötvös Loránd University, Egyetem tér 1–3, H-1053, Budapest, Hungary

    Péter Komjáth

Authors
  1. Nathan Bowler
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  2. Johannes Carmesin
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  3. Péter Komjáth
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  4. Christian Reiher
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Corresponding authors

Correspondence to Nathan Bowler, Johannes Carmesin, 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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  • DOI: https://doi.org/10.1007/s00493-019-4045-9

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