Sum-free sets in loops

Abstract

The examples in section 4 show how, given a colouring of a graph, it may be possible to find a sum-free partition of a loop yielding that colouring; the converse problem, of finding a sum-free partition and then obtaining the colouring looks like being much harder. Note also that there are still colourings of K₆ which cannot be obtained in this way. For example, if the colouring shown in Figure 7 were obtainabl from a loop {0,1,2,3,4,5} then three elements, say {1,2,3} would have to be in one of the sum-free sets. But then {0,4,5} would, by 3.1, be a subloop, and so neither 0–4 nor 4–0 could belong to {1,2,3}. (However, Heinrich [4] has shown that this colouring can be embedded in a colouring of K₁₀ obtained from a sum-free partition of Z₁₀).