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The probability that a random multigraph is simple

Svante Janson
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Abstract

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.

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Svante Janson. 2006-09-28. The probability that a random multigraph is simple. https://arxiv.org/abs/math/0609802

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