### 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 set D of prescribed differences is a proper subset of the primes. In particular, we prove that if D 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 sets D which necessitate four colours remains open.

Original language | English |
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Pages (from-to) | 17-32 |

Number of pages | 16 |

Journal | Graphs and Combinatorics |

Volume | 6 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 1990 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Eggleton, R. B., Erdös, P., & Skilton, D. K. (1990). Colouring prime distance graphs.

*Graphs and Combinatorics*,*6*(1), 17-32. https://doi.org/10.1007/BF01787476