Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Annika Heckel

TL;DR

This work addresses Erdős and Gimbel's question on whether the usual chromatic number $\chi(G)$ and the cochromatic number $\zeta(G)$ of a random graph $G\sim G_{n,1/2}$ diverge in the limit. By transferring lower bounds for $\chi$ from recent colorability results to the cochromatic setting, and by a refined second-moment analysis over structured color/ cocolor profiles (including a carefully chosen tame profile $\mathbf{k}^*$), the authors show that whp $\chi(G)-\zeta(G)\ge n^{1-\varepsilon}$ for roughly 95% of values of $n$ satisfying a growth condition on the expected number of independent sets of size $\alpha$. The approach hinges on defining $t$-bounded colourings, analyzing both colourings and cocolorings via profile counts, and utilizing Paley–Zygmund together with Azuma–Hoeffding to upgrade probabilistic bounds to high-probability statements. The results illuminate the distinct concentration behavior of $\chi$ and $\zeta$ in random graphs and lay groundwork toward a full resolution, with conjectures suggesting a tighter scale $\Theta(n/\log^3 n)$ for all $n$.

Abstract

The cochromatic number $ζ(G)$ 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 $χ(G)$ denote the usual chromatic number. Around 1991 Erdős and Gimbel asked: For the random graph $G \sim G_{n, 1/2}$, does $χ(G)-ζ(G) \rightarrow \infty$ whp? Erdős 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 $n$.

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

TL;DR

This work addresses Erdős and Gimbel's question on whether the usual chromatic number and the cochromatic number of a random graph diverge in the limit. By transferring lower bounds for from recent colorability results to the cochromatic setting, and by a refined second-moment analysis over structured color/ cocolor profiles (including a carefully chosen tame profile ), the authors show that whp for roughly 95% of values of satisfying a growth condition on the expected number of independent sets of size . The approach hinges on defining -bounded colourings, analyzing both colourings and cocolorings via profile counts, and utilizing Paley–Zygmund together with Azuma–Hoeffding to upgrade probabilistic bounds to high-probability statements. The results illuminate the distinct concentration behavior of and in random graphs and lay groundwork toward a full resolution, with conjectures suggesting a tighter scale for all .

Abstract

The cochromatic number of a graph is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let denote the usual chromatic number. Around 1991 Erdős and Gimbel asked: For the random graph , does whp? Erdős offered \1,000 for a negative answer. We give a positive answer to this question for roughly 95% of all values .
Paper Structure (18 sections, 15 theorems, 72 equations)

This paper contains 18 sections, 15 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Setup and notation
  3. Background on the independence number of $G_{n, 1/2}$
  4. (Co-)colouring profiles and the $t$-bounded chromatic number
  5. Counts of colourings and co-colourings
  6. Technical lemma
  7. Proof of Theorem \ref{['theorem:difference']}
  8. Proof strategy
  9. The lower bound on $\chi(G_{n, 1/2})$
  10. The upper bound on $\zeta({G_{n, 1/2}})$
  11. Proof of Proposition \ref{['proposition:main']}
  12. Tame profiles
  13. The random variables $Z_k$ and $Z^{\mathrm{co}}_k$
  14. Relevant pairs
  15. Bounding the second moment
  16. ...and 3 more sections

Key Result

Theorem 1

Fix $\varepsilon>0$, and let $n$ be such that $n^{0.05+\varepsilon}\leqslant \mu_\alpha \leqslant n^{1-\varepsilon}$. Let $G \sim {G_{n, 1/2}}$, then whp,

Theorems & Definitions (21)

  • Theorem 1
  • Lemma 2: heckel2023colouring, Le. 3.7
  • Lemma 3: heckel2023colouring, Lemma 8.1
  • Corollary 4
  • Proposition 5
  • Proposition 6
  • proof
  • Definition 7: heckel2023colouring, Def. 2.3
  • Definition 8: heckel2023colouring, Def. 4.1
  • Definition 9
  • ...and 11 more