The probability that a random multigraph is simple, II

TL;DR

This paper provides a new proof using the method of moments for the asymptotic probability that a random multigraph with given degrees is simple, showing it stays away from zero under specific degree sum conditions.

Contribution

It introduces a novel proof technique for the probability of simplicity in random multigraphs, applicable to bipartite graphs and convergence in distribution.

Findings

Probability remains bounded away from zero if and only if degree sum condition holds.

New proof method using the method of moments.

Includes results for bipartite graphs.

Abstract

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.