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.

Source : oai:HAL:hal-00958983v1

Volume: Vol. 5

Published on: January 1, 2002

Submitted on: March 26, 2015

Keywords: thresholds,sparse random graphs,3-colourability,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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