The probability that a random multigraph is simple
Abstract
Consider a random multigraph G* with given vertex degrees d1,...,dn, contructed by the configuration model. We show that, asymptotically for a sequence of such multigraphs with the number of edges (d1+...+dn)/2 tending to infinity, the probability that the multigraph is simple stays away from 0 if and only if Σ di2=O(Σ di). 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.
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