Counting partitions of $ G_{n,1/2} $ with degree congruence conditions - Balister, P; Powierski, E; Scott, A; Tan, J (Wiley, 2022)
For $G=G_{n, 1/2}$, the Erdős--Renyi random graph, let $X_n$ be the random variable representing the number of distinct partitions of $V(G)$ into sets $A_1, …, A_q$ so that the degree of each vertex in $G[A_i]$ is divisible by $q$ for all $i\in[q]$. We prove that if $q≥ 3$ is odd then $X_n\xrightarrow{d}{Po(1/q!)}$, and if $q ≥ 4$ is even then $X_n\xrightarrow{d}{Po(2^q/q!)}$. More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in $G[A_i]$ to be congruent to $x_i$ modulo $q$ for each $i\in[q]$, where the residues $x_i$ may be chosen freely. For $q=2$, the distribution is not asymptotically Poisson, but it can be determined explicitly.