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Erdős #1032 · graph theory / chromatic number

We say that a graph is 44-chromatic critical if it has chromatic number 44, and removing any edge decreases the chromatic number to 33.

Is there, for arbitrarily large nn, a 44-chromatic critical graph on nn verti…

MSC 05C · Graph theory

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Problem statement

We say that a graph is 44-chromatic critical if it has chromatic number 44, and removing any edge decreases the chromatic number to 33.

Is there, for arbitrarily large nn, a 44-chromatic critical graph on nn vertices with minimum degree ≫n\gg n?

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Background

In [Er93] Erdős said he asked this 'more than 20 years ago'.

Dirac gave an example of a 66-chromatic critical graph with minimum degree >n/2>n/2. This problem is also open for 55-chromatic critical graphs.

Simonovits [Si72] and Toft [To72] independently constructed 44-chromatic critical graphs with minimum degree ≫n1/3\gg n^{1/3}. Toft conjectured that a 44-chromatic critical graph on nn vertices has at least (53+o(1))n(\frac{5}{3}+o(1))n vertices, and has examples to show this would be the best possible.

See also [917] and [944].

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The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1032 link

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