Reed's recolouring conjecture

Let $G$ be a graph, with clique number $4\omega(G)4$ and maximum degree $4\Delta(G)4$. A proper $k$-colouring is Kempe-recolourable when it can be transformed through Kempe changes, which interchange two colours on a connected component of the subgraph induced by those colours.

Reed's recolouring conjecture. Any graph $G$ is $k$-recolourable for all

$$k \geqslant \left\lceil \frac{1}{3}\omega(G) + \frac{2}{3}(\Delta(G) +1) \right\rceil.$$

The conjecture is a recolouring analogue of Reed's conjecture that the chromatic number is at most $\left\lceil(\omega(G)+\Delta(G)+1)/2\right\rceil$. The paper proves that frozen colourings are unique up to permuting colours below this threshold, while constructions show the bound is tight; the recolouring assertion remains open.