  We investigate the distribution of monochromatic subgraph counts in random
vertex $2$-colorings of large graphs. We give sufficient conditions for the
asymptotic normality of these counts and demonstrate their essential necessity
(particularly for monochromatic triangles). Our approach refines the
fourth-moment theorem to establish new, local influence-based conditions for
asymptotic normality; these findings more generally provide insight into
fourth-moment phenomena for a broader class of Rademacher and Gaussian
polynomials.
