A fourth‐moment phenomenon for asymptotic normality of monochromatic subgraphs

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