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A non-imbeddable proper colouring

  • Katherine Heinrich
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 560)

Abstract

To a sum-free partition of a group of order n into r parts, there corresponds a triangle-free edge-colouring of the complete graph on n vertices into r colours. We say that this colouring, and all the complete subgraphs of it, are derived from the sum-free partition.

It has been asked whether every triangle-free colouring of a complete graph in r colours can be derived from some sum-free partition into r parts. We prove that the answer is "no", by exhibiting a triangle-free colouring of K6 into three colours which cannot be imbedded in any colouring derived from a sum-free partition into three parts.

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References

  1. [1]
    Anne Penfold Street, Embedding proper colourings. These proceedings.Google Scholar
  2. [2]
    Anne Penfold Street and W.D. Wallis, Sum-free sets, coloured graphs and designs, J. Austral. Math. Soc. (to appear).Google Scholar
  3. [3]
    W.D. Wallis, Anne Penfold Street and Jennifer Seberry Wallis, Cominatorics: Room Squares, Sum-free Sets, Hadamard Matrices. Lecture Notes in Mathematics 292, Springer-Verlag, Berlin, Heidelberg, New York, 1972.CrossRefGoogle Scholar
  4. [4]
    E.G. Whitehead Jr., Difference sets and sum-free sets in groups of order 16. Discrete Math. 13 (1975), 399–407.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • Katherine Heinrich

There are no affiliations available

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