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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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    de Bruijn, N.G., Erdös, P.: A colour problem for infinite graphs and a problem in the theory of relations. Indagationes Math.13, 371–373 (1951)

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    Eggleton, R.B., Erdös, P., Skilton, D.K.: Colouring the real line, J. Comb. Theory (B)39, 86–100 (1985)

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    Eggleton, R.B., Erdös P., Skilton, D.K.: Research Problem 77, Discrete Math.58, 323 (1986)

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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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Keywords

  • Proper Subset
  • Distance Graph
  • Prime Distance
  • Periodic Colouring
  • Prescribe Difference