Chen--Lih--Wu's equitable Brooks conjecture

Let $G$ be a connected finite graph of maximum degree $\Delta \geqslant 1$. An equitable $\Delta$-coloring is a proper coloring with $\Delta$ color classes whose sizes differ by at most $1$.

Chen--Lih--Wu's conjecture. The graph $G$ has an equitable $\Delta$-coloring unless one of the following holds: $\Delta=2$ and $G$ is an odd cycle; $G \cong K_{\Delta+1}$; or $\Delta$ is odd and $G \cong K_{\Delta,\Delta}$.

This is an equitable analogue of Brooks's theorem. The conjecture remains open, and the paper studies measurable and Borel versions of related coloring results.