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

Abstract

AbstractFor , the Erdős–Renyi random graph, let be the random variable representing the number of distinct partitions of into sets so that the degree of each vertex in is divisible by for all . We prove that if is odd then , and if is even then . More generally, we show that the distribution is still asymptotically Poisson when we require all degrees in to be congruent to modulo for each , where the residues may be chosen freely. For , the distribution is not asymptotically Poisson, but it can be determined explicitly.