The probability that a random multigraph is simple, II

Consider a random multigraph with given vertex degrees constructed by the configuration model. We give a new proof of the fact that, asymptotically for a sequence of such multigraphs with the number of edges tending to infinity, the probability that the multigraph is simple stays away from 0 if and only if $\sum d_i^2 = O(\sum d_i)$, where $d_i$ are the vertex degrees. The new proof uses the method of moments, which makes it possible to use it in some applications concerning convergence in distribution. Corresponding results for bipartite graphs are included.