  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.
