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

Nikolaos Fountoulakis (1) , Colin Mcdiarmid (2)

1 MI - Mathematical Institute [Oxford]
2 Department of Statistics [Oxford]

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.

Mots clés: sparse random graphs; 3-colourability; thresholds

Domaines: Mathématique discrète [cs.DM]

Nikolaos Fountoulakis, Colin Mcdiarmid. Upper bounds on the non-3-colourability threshold of random graphs. Discrete Mathematics and Theoretical Computer Science, 2002, Vol. 5, pp.205-226. ⟨10.46298/dmtcs.299⟩. ⟨hal-00958983⟩