On the number of solutions in random graph $k$-colouring

TL;DR

This paper precisely characterizes the asymptotic distribution of the logarithm of the number of proper k-colourings in random graphs, linking fluctuations to small cycle counts, across a broad degree range.

Contribution

It provides the first exact asymptotic distribution of the number of k-colourings in random graphs, extending to degrees up to the condensation phase transition for large k.

Findings

Distribution of ln Z_k(G(n,m)) is determined asymptotically.

Fluctuations are linked to small cycle counts.

Results apply to a wide range of average degrees.

Abstract

Let $k \ge 3$ be a fixed integer. We exactly determine the asymptotic distribution of $\ln Z_k(G(n,m))$, where $Z_k(G(n,m))$ is the number of $k$-colourings of the random graph $G(n,m)$. A crucial observation to this aim is that the fluctuations in the number of colourings can be attributed to the fluctuations in the number of small cycles in $G(n,m)$. Our result holds for a wide range of average degrees, and for $k$ exceeding a certain constant $k_0$ it covers all average degrees up to the so-called "condensation phase transition".