The difference between the chromatic and the cochromatic number of a random graph

TL;DR

This paper investigates the difference between the chromatic number and the cochromatic number in random graphs, showing that for most sizes, this difference tends to infinity, answering a long-standing question posed by Erdős and Gimbel.

Contribution

It provides a positive answer to Erdős and Gimbel's question, demonstrating that the difference between the chromatic and cochromatic numbers diverges for most random graphs.

Findings

The difference between $ ext{chi}(G)$ and $ ext{zeta}(G)$ tends to infinity for most $n$.

Confirms Erdős and Gimbel's conjecture for approximately 95% of graph sizes.

Addresses a question with implications for graph coloring theory.

Abstract

The cochromatic number $\zeta(G)$ of a graph $G$ is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let $\chi(G)$ denote the usual chromatic number. Around 1991 Erd\H{o}s and Gimbel asked: For the random graph $G \sim G_{n, 1/2}$, does $\chi(G)-\zeta(G) \rightarrow \infty$ whp? Erd\H{o}s offered \$100 for a positive and \$1,000 for a negative answer. We give a positive answer to this question for roughly 95% of all values $n$.