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Graphs and Combinatorics

, Volume 6, Issue 1, pp 17–32 | Cite as

Colouring prime distance graphs

  • R. B. Eggleton
  • P. Erdös
  • D. K. Skilton
Article

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.

Keywords

Proper Subset Distance Graph Prime Distance Periodic Colouring Prescribe Difference 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    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)Google Scholar
  2. 2.
    Eggleton, R.B., Erdös, P., Skilton, D.K.: Colouring the real line, J. Comb. Theory (B)39, 86–100 (1985)Google Scholar
  3. 3.
    Eggleton, R.B., Erdös P., Skilton, D.K.: Research Problem 77, Discrete Math.58, 323 (1986)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • R. B. Eggleton
    • 1
  • P. Erdös
    • 2
  • D. K. Skilton
    • 3
  1. 1.Department of MathematicsUniversity of Brunei DarussalamGadongBrunei Darussalam
  2. 2.Hungarian Academy of SciencesBudapest VHungary
  3. 3.IBM AustraliaUltimoAustralia

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