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

Abstract

For G=Gn,1/2$$ G={G}_{n,1/2} $$ , the Erdős–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,…,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∈[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∈[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.