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Upper bounds on the non-3-colourability threshold of random graphs

Abstract : We present a full analysis of the expected number of 'rigid' 3-colourings of a sparse random graph. This shows that, if the average degree is at least 4.99, then as n → ∞ the expected number of such colourings tends to 0 and so the probability that the graph is 3-colourable tends to 0. (This result is tight, in that with average degree 4.989 the expected number tends to ∞.) This bound appears independently in Kaporis \textitet al. [Kap]. We then give a minor improvement, showing that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.
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Nikolaos Fountoulakis, Colin Mcdiarmid. Upper bounds on the non-3-colourability threshold of random graphs. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2002, 5, pp.205-226. ⟨hal-00958983⟩

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