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
Anne Penfold Street, Embedding proper colourings. These proceedings.
Anne Penfold Street and W.D. Wallis, Sum-free sets, coloured graphs and designs, J. Austral. Math. Soc. (to appear).
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.
E.G. Whitehead Jr., Difference sets and sum-free sets in groups of order 16. Discrete Math. 13 (1975), 399–407.
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© 1976 Springer-Verlag
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Heinrich, K. (1976). A non-imbeddable proper colouring. In: Casse, L.R.A., Wallis, W.D. (eds) Combinatorial Mathematics IV. Lecture Notes in Mathematics, vol 560. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097371
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DOI: https://doi.org/10.1007/BFb0097371
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