A fourth moment phenomenon for asymptotic normality of monochromatic subgraphs

TL;DR

This paper establishes a fourth moment phenomenon for the asymptotic normality of the count of monochromatic subgraphs in random graph colorings, providing explicit error rates and conditions for convergence to normality.

Contribution

It proves a fourth moment characterization for the normal limit of monochromatic subgraph counts in random graph colorings, with explicit error bounds.

Findings

Normal approximation for monochromatic subgraph counts

Fourth moment convergence characterizes normality for large color counts

Explicit error rates in the central limit theorem

Abstract

Given a graph sequence $\{G_n\}_{n\ge1}$ and a simple connected subgraph $H$, we denote by $T(H,G_n)$ the number of monochromatic copies of $H$ in a uniformly random vertex coloring of $G_n$ with $c \ge 2$ colors. In this article, we prove a central limit theorem for $T(H,G_n)$ with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of $H$ that we call good joins. Counts of good joins are closely related to the fourth moment of a normalized version of $T(H,G_{n})$, and that connection allows us to show a fourth moment phenomenon for the central limit theorem. Precisely, for $c\ge 30$, we show that $T(H,G_n)$ (appropriately centered and rescaled) converges in distribution to $\mathcal{N}(0,1)$ whenever its fourth moment converges to 3 (the fourth moment of the standard normal distribution). We show the convergence of the fourth moment is necessary to obtain a normal limit when $c\ge 2$. The combination of these results implies that the fourth moment condition characterizes the limiting normal distribution of $T(H,G_n)$ for all subgraphs $H$, whenever $c\ge 30$.