The probability that a random multigraph is simple. II

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.

[1]  Brendan D. McKay,et al.  Asymptotic enumeration by degree sequence of graphs with degreeso(n1/2) , 1991, Comb..

[2]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[3]  Brendan D. McKay,et al.  Uniform Generation of Random Regular Graphs of Moderate Degree , 1990, J. Algorithms.

[4]  Nicholas C. Wormald,et al.  The asymptotic distribution of short cycles in random regular graphs , 1981, J. Comb. Theory, Ser. B.

[5]  Svante Janson,et al.  Asymptotic normality of the k-core in random graphs , 2008 .

[6]  Svante Janson.  The Probability That a Random Multigraph is Simple , 2009, Comb. Probab. Comput..

[7]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[8]  Alan M. Frieze,et al.  The Size of the Largest Strongly Connected Component of a Random Digraph with a Given Degree Sequence , 2004, Combinatorics, Probability and Computing.

[9]  Béla Bollobás,et al.  An old approach to the giant component problem , 2012, J. Comb. Theory, Ser. B.

[10]  Colin Cooper,et al.  The cores of random hypergraphs with a given degree sequence , 2004, Random Struct. Algorithms.

[11]  Brendan D. McKay,et al.  Asymptotic Enumeration by Degree Sequence of Graphs of High Degree , 1990, Eur. J. Comb..

[12]  N. Wormald.  Some problems in the enumeration of labelled graphs , 1980, Bulletin of the Australian Mathematical Society.

[13]  Jose H. Blanchet,et al.  Characterizing optimal sampling of binary contingency tables via the configuration model , 2010, Random Struct. Algorithms.

[14]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[15]  Svante Janson,et al.  Law of large numbers for the SIR epidemic on a random graph with given degrees , 2013, Random Struct. Algorithms.

[16]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[17]  A. Gut.  Probability: A Graduate Course , 2005 .

[18]  Brendan D. McKay,et al.  Asymptotic enumeration of sparse 0-1 matrices with irregular row and column sums , 2006, J. Comb. Theory, Ser. A.

[19]  Béla Bollobás,et al.  Random Graphs , 1985 .