On a question of Erd\H{o}s and Gimbel on the cochromatic number

TL;DR

This paper investigates the relationship between chromatic and cochromatic numbers in random graphs, showing that their difference can grow significantly, thus answering a question posed by Erdős and Gimbel.

Contribution

It demonstrates that the difference between chromatic and cochromatic numbers in G_{n,1/2} is not bounded by n^{1/2 - o(1)} with high probability, providing new insights into graph coloring properties.

Findings

The difference between chromatic and cochromatic numbers can be arbitrarily large.

Addresses a longstanding open question by Erdős and Gimbel.

Provides probabilistic bounds on graph coloring parameters.

Abstract

In this note, we show that the difference between the chromatic and the cochromatic number of the random graph $G_{n,1/2}$ is not whp bounded by $n^{1/2-o(1)}$, addressing a question of Erd\H{o}s and Gimbel.