The Colouring Number of Infinite Graphs


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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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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Corresponding authors

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).

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Mathematics Subject Classification (2010)

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