On the difference between the chromatic and cochromatic number

The cochromatic number $\zeta (G)$ of a graph $G$ is the smallest number of colors in a vertex-coloring of $G$ such that every color class forms an independent set or a clique. In three papers written around 1990, Erdős, Gimbel and collaborators raised several open problems regarding the relationship of the chromatic and cochromatic number of a graph. In this short note, we address several of these problems, in particular -we disprove a conjecture of Erdős, Gimbel and Straight from 1988, -answer negatively a problem posed by Erdős and Gimbel in 1993, and -give positive evidence for a 1000$–question of Erdős and Gimbel.