Frozen $(\Delta+1)$-colourings of bounded degree graphs

Let $G$ be a graph of maximum degree $\Delta$ and $k$ be an integer. The $k$-recolouring graph of $G$ is the graph whose vertices are $k$-colourings of $G$ and where two $k$-colourings are adjacent if they differ at exactly one vertex. It is well-known that the $k$-recolouring graph is connected for $k\geq \Delta+2$. Feghali, Johnson and Paulusma [Journal of Graph Theory, 83(4):340--358] showed that the $(\Delta+1)$-recolouring graph is composed by a unique connected component of size at least $2$ and (possibly many) isolated vertices. In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the $(\Delta+1)$-recolouring graph. Our main contribution is to show that, if $G$ is connected, the proportion of frozen colourings of $G$ is exponentially smaller than the total number of colourings. This motivates the study of the Glauber dynamics on $(\Delta+1)$-colourings. In contrast to the conjectured mixing time for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for $(\Delta+1)$-colourings can be of quadratic order. Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.