On the Number of Solutions in Random Graphk-Colouring

Abstract

Letk⩾ 3 be a fixed integer. We exactly determine the asymptotic distribution of lnZk(G(n, m)), whereZk(G(n, m)) is the number ofk-colourings of the random graphG(n, m). A crucial observation to this end is that the fluctuations in the number of colourings can be attributed to the fluctuations in the number of small cycles inG(n, m). Our result holds for a wide range of average degrees, and forkexceeding a certain constantk0it covers all average degrees up to the so-calledcondensation phase transition.