Graph decomposition and parity

Motivated by a recent extension of the zero-one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph Gn,p. We use an idea of graph decompositions to give a sufficient condition for this distribution to tend to uniform modulo q. We determine the asymptotic distribution of all fixed two-component graphs in Gn,p for all q, and we give infinite families of many-component graphs with a uniform asymptotic distribution for all q. We also prove a negative result that no recursive proof of the simplest form exists for a uniform asymptotic distribution for arbitrary graphs.