The Probability That a Random Multigraph is Simple

Consider a random multigraph G* with given vertex degrees d(1),...,d(n), constructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges 1/2 Sigma(i) d(i) -> infinity, the probability that the multigraph is simple stays away from 0 if and only if Sigma(i) d(i)(2) = O (Sigma(i) d(i)). This was previously known only under extra assumptions on the maximum degree max(i) d(i). We also give an asymptotic formula for this probability, extending previous results by several authors.