A universal error bound in the CLT for counting monochromatic edges in uniformly colored graphs

TL;DR

This paper establishes a universal error bound for the central limit theorem concerning the count of monochromatic edges in randomly colored graphs, using Stein's method to improve upon previous results that lacked convergence rates.

Contribution

The authors apply Stein's method to derive a universal error bound for the CLT of monochromatic edges, independent of graph structure, advancing the understanding of convergence rates.

Findings

Universal error bound depending only on edges and colors

Application of Stein's method to graph colorings

Improved CLT convergence rate results

Abstract

Let $\{G_n: n\geq 1\}$ be a sequence of simple graphs. Suppose $G_n$ has $m_n$ edges and each vertex of $G_n$ is colored independently and uniformly at random with $c_n$ colors. Recently, Bhattacharya, Diaconis and Mukherjee (2013) proved universal limit theorems for the number of monochromatic edges in $G_n$. Their proof was by the method of moments, and therefore was not able to produce rates of convergence. By a non-trivial application of Stein's method, we prove that there exists a universal error bound for their central limit theorem. The error bound depends only on $m_n$ and $c_n$, regardless of the graph structure.