Counting partitions of Gn,1/2$$ {G}_{n,1/2} $$ with degree congruence conditions

For G=Gn,1/2$$ G={G}_{n,1/2} $$, the Erdos-Renyi random graph, let Xn$$ {X}_n $$ be the random variable representing the number of distinct partitions of V(G)$$ V(G) $$ into sets A1, horizontal ellipsis ,Aq$$ {A}_1,\dots, {A}_q $$ so that the degree of each vertex in G[Ai]$$ G\left[{A}_i\right] $$ is divisible by q$$ q $$ for all i is an element of[q]$$ i\in \left[q\right] $$. We prove that if q >= 3$$ q\ge 3 $$ is odd then Xn -> dPo(1/q!)$$ {X}_n\overset{d}{\to \limits}\mathrm{Po}\left(1/q!\right) $$, and if q >= 4$$ q\ge 4 $$ is even then Xn -> dPo(2q/q!)$$ {X}_n\overset{d}{\to \limits}\mathrm{Po}\left({2}<^>q/q!\right) $$. More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in G[Ai]$$ G\left[{A}_i\right] $$ to be congruent to xi$$ {x}_i $$ modulo q$$ q $$ for each i is an element of[q]$$ i\in \left[q\right] $$, where the residues xi$$ {x}_i $$ may be chosen freely. For q=2$$ q=2 $$, the distribution is not asymptotically Poisson, but it can be determined explicitly.