Frozen (Δ+1)-colourings of bounded degree graphs
Résumé
Let G be a graph of maximum degree Δ 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≥Δ+2. Feghali, Johnson and Paulusma [Journal of Graph Theory, 83(4):340--358] showed that the (Δ+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 (Δ+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 (Δ+1)-colourings. In contrast to the conjectured mixing time for k≥Δ+2 colours, we show that the mixing time of the Glauber dynamics for (Δ+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.