A fourth-moment phenomenon for asymptotic normality of monochromatic subgraphs

Given a graph sequence $$$ {\left\{{G}_n\right\}}_{n\ge 1} $$$ and a simple connected subgraph $$$ H $$$, we denote by $$$ T\left(H,{G}_n\right) $$$ the number of monochromatic copies of $$$ H $$$ in a uniformly random vertex coloring of $$$ {G}_n $$$ with $$$ c\ge 2 $$$ colors. We prove a central limit theorem for $$$ T\left(H,{G}_n\right) $$$ (we denote the appropriately centered and rescaled statistic as $$$ Z\left(H,{G}_n\right) $$$) with explicit error rates. The error rates arise from graph counts of collections formed by joining copies of $$$ H $$$ which we call good joins. Good joins are closely related to the fourth moment of $$$ Z\left(H,{G}_n\right) $$$, which allows us to show a fourth moment phenomenon for the central limit theorem. For $$$ c\ge 30 $$$, we show that $$$ Z\left(H,{G}_n\right) $$$ converges in distribution to $𝒩(0,1)$ whenever its fourth moment converges to 3. We show the convergence of the fourth moment is necessary to obtain a normal limit when $$$ c\ge 2 $$$.