Upper bounds on the non-3-colourability threshold of random graphs
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-->infinity 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 infinity.) This bound appears independently in Kaporis et al. [14]. We then give a minor improvement, which shows in particular that the probability that the graph is 3-colourable tends to 0 if the average degree is at least 4.989.
3-colourability thresholds sparse random graphs