Counting partitions of $G_{n,1/2}$ with degree congruence conditions

For G = Gn,1/2, the Erdős–Renyi random graph, let Xn be the random variable representing the number of distinct partitions of V (G) into sets A1, . . . , Aq so that the degree of each vertex in G[Ai] is divisible by q for all i ∈ [q]. We prove that if q ≥ 3 is odd then Xn d −→ Po(1/q!), and if q ≥ 4 is even then Xn d −→ Po(2q/q!). More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G[Ai] to be congruent to xi modulo q for each i ∈ [q], where the residues xi may be chosen freely. For q = 2, the distribution is not asymptotically Poisson, but it can be determined explicitly.