  Consider a random multigraph G* with given vertex degrees d_1,...,d_n,
contructed by the configuration model. We show that, asymptotically for a
sequence of such multigraphs with the number of edges (d_1+...+d_n)/2 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). This was previously known only under extra
assumtions on the maximum degree. We also give an asymptotic formula for this
probability, extending previous results by several authors.
