2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/116698Consider 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.24 pagesCombinatoricsProbability05C80; 05C30, 60C05text