A non-imbeddable proper colouring
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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