On the difference between the chromatic and cochromatic number

TL;DR

This paper investigates the relationship between the chromatic and cochromatic numbers of graphs, disproving some longstanding conjectures, answering open problems negatively, and providing positive evidence for a related question.

Contribution

It resolves several open problems from the 1990s regarding chromatic and cochromatic numbers, including disproving a conjecture and answering a posed question.

Findings

Disproved a conjecture of Erdős, Gimbel, and Straight from 1988.

Answered negatively a problem posed by Erdős and Gimbel in 1993.

Provided positive evidence for a 1000-dollar question of Erdős and Gimbel.

Abstract

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, Erd\H{o}s, 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 Erd\H{o}s, Gimbel and Straight from 1988, -answer negatively a problem posed by Erd\H{o}s and Gimbel in 1993, and -give positive evidence for a 1000\$--question of Erd\H{o}s and Gimbel.