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

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.