Meyer's equitable coloring conjecture

Let $G$ be a connected graph, and let

\Delta$(G)$ denote its maximum degree. An **equitable $r$-coloring** is a proper coloring of $G$ with $r$ colors whose color classes differ in size by at most $1$. **Meyer's equitable coloring conjecture.** If $G$ is connected, but not a complete graph or an odd cycle, then $G$ has an equitable

Delta$(G)$-coloring.

This conjecture is an equitable analogue of Brooks' theorem and concerns the equitable chromatic number. The paper's abstract proves the stronger-looking result for 1-planar graphs with maximum degree at most $r$ when $r\ge13$; no resolution of this general conjecture is supplied in the given context.