Planting Colourings Silently

Let k ≥ 3 be a fixed integer and let Zk(G) be the number of k-colourings of the graph G. For certain values of the average degree, the random variable Zk(G(n,m)) is known to be concentrated in the sense that 1 n (lnZk(G(n,m)) − lnE[Zk(G(n,m))]) converges to 0 in probability [Achlioptas and Coja-Oghlan: Proc. FOCS 2008]. In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees, 1 ω (lnZk(G(n,m)) − lnE[Zk(G(n,m))]) converges to 0 in probability for any diverging function ω = ω(n) → ∞. For k exceeding a certain constant k0 this result covers all average degrees up to the so-called condensation phase transition dk,cond, and this is best possible. As an application, we show that the experiment of choosing a k-colouring of the random graph G(n,m) uniformly at random is contiguous with respect to the so-called “planted model”.