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On a question of Erdős and Gimbel on the cochromatic number

Annika Heckel

TL;DR

The paper addresses Erdős and Gimbel's question on whether the gap between the chromatic number $\chi(G)$ and the cochromatic number $\zeta(G)$ in the random graph $G_{n,1/2}$ tends to infinity. It proves that this difference is not whp bounded by $n^{1/2-o(1)}$, tying any such bound to the concentration interval length of $\chi(G_{n,1/2})$ through a comparison with the complement graph and Harris's lemma, and employing non-concentration results for $\chi(G_{n,1/2})$. A key proposition shows that any proposed bound $g(n)$ with $\mathbb{P}(\chi(G)-\zeta(G)\le g(n))>0.999$ must eventually satisfy $g(n) > c \frac{\sqrt{n}\log\log n}{\log^3 n}$ along a subsequence, highlighting the link to chromatic concentration. The author conjectures whp $\chi(G)-\zeta(G)=\Theta(n/\log^3 n)$, which aligns with a first-moment heuristic for the thresholds of colorings vs cocolorings in $G_{n,1/2}$.

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 Erdős and Gimbel.

On a question of Erdős and Gimbel on the cochromatic number

TL;DR

The paper addresses Erdős and Gimbel's question on whether the gap between the chromatic number and the cochromatic number in the random graph tends to infinity. It proves that this difference is not whp bounded by , tying any such bound to the concentration interval length of through a comparison with the complement graph and Harris's lemma, and employing non-concentration results for . A key proposition shows that any proposed bound with must eventually satisfy along a subsequence, highlighting the link to chromatic concentration. The author conjectures whp , which aligns with a first-moment heuristic for the thresholds of colorings vs cocolorings in .

Abstract

In this note, we show that the difference between the chromatic and the cochromatic number of the random graph is not whp bounded by , addressing a question of Erdős and Gimbel.
Paper Structure (3 sections, 3 theorems, 12 equations)

This paper contains 3 sections, 3 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['theorem:cochromatic']}
  3. Discussion

Key Result

Theorem 1

Let $G \sim G_{n, 1/2}$. There is a constant $c>0$ so that for any sequence of integers $g(n)$ such that there is a sequence of integers $n^*$ such that

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2: HRHowdoesheckel2023colouring
  • Proposition 3
  • proof : Proof of Proposition \ref{['prop:hilfs']}
  • Conjecture 4