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

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