  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 induces an
independent set or a clique. In 1988, Erd\H{o}s, Gimbel and Straight
conjectured that every $K_5$-free graph $G$ with $\zeta(G)>3$ satisfies
$\chi(G)\le \zeta(G)+2$. In this note, we present a counterexample to this
conjecture and discuss related results and questions.
