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Colouring prime distance graphs

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Abstract

Four colours are necessary and sufficient to colour all the integers so that any two with difference equal to a prime have different colours. We investigate the corresponding problem when the setD of prescribed differences is a proper subset of the primes. In particular, we prove that ifD contains {2, 3} and also contains a pair of twin primes (one of which may be 3), then four colours are necessary. Numerous results regarding periodic colourings are also obtained. However, the problem of characterizing those setsD which necessitate four colours remains open.

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References

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Authors and Affiliations

  1. Department of Mathematics, University of Brunei Darussalam, 3186, Gadong, B.S.B., Brunei Darussalam

    R. B. Eggleton

  2. Hungarian Academy of Sciences, Reáltanoda U. 13-15, Budapest V, Hungary

    P. Erdös

  3. IBM Australia, 37 Ada Place, 2007, Ultimo, N.S.W., Australia

    D. K. Skilton

Authors
  1. R. B. Eggleton
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  2. P. Erdös
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  3. D. K. Skilton
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Eggleton, R.B., Erdös, P. & Skilton, D.K. Colouring prime distance graphs. Graphs and Combinatorics 6, 17–32 (1990). https://doi.org/10.1007/BF01787476

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  • DOI: https://doi.org/10.1007/BF01787476

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