## Journal of Applied Probability

### The probability that a random multigraph is simple. II

Svante Janson

#### Abstract

Consider a random multigraph G* with given vertex degrees d1, . . ., dn, 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 ½∑idi → ∞, the probability that the multigraph is simple stays away from 0 if and only if ∑idi = O(∑idi). 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.

#### Article information

Source
J. Appl. Probab., Volume 51A (2014), 123-137.

Dates
First available in Project Euclid: 2 December 2014

https://projecteuclid.org/euclid.jap/1417528471

Digital Object Identifier
doi:10.1239/jap/1417528471

Mathematical Reviews number (MathSciNet)
MR3317354

Zentralblatt MATH identifier
1309.05162

#### Citation

Janson, Svante. The probability that a random multigraph is simple. II. J. Appl. Probab. 51A (2014), 123--137. doi:10.1239/jap/1417528471. https://projecteuclid.org/euclid.jap/1417528471

#### References

• Békéssy, A., Békéssy, P. and Komlós, J. (1972). Asymptotic enumeration of regular matrices. Studia Sci. Math. Hungar. 7, 343–353.
• Bender, E. A. and Canfield, E. R. (1978). The asymptotic number of labeled graphs with given degree sequences. J. Combinatorial Theory A 24, 296–307.
• Blanchet, J. and Stauffer, A. (2013). Characterizing optimal sampling of binary contingency tables via the configuration model. Random Structures Algorithms 42, 159–184.
• Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combinatorics 1, 311–316.
• Bollobás, B. (2001). Random Graphs, 2nd edn. Cambridge University Press.
• Bollobás, B. and Riordan, O. (2012). An old approach to the giant component problem. Preprint. Available at http://arxiv.org/abs/1209.3691v1.
• Flajolet, P. and Sedgewick, R. (2009). Analytic Combinatorics. Cambridge University Press.
• Greenhill, C., McKay, B. D. and Wang, X. (2006). Asymptotic enumeration of sparse $0$-$1$ matrices with irregular row and column sums. J. Combinatorial Theory A 113, 291–324.
• Gut, A. (2013). Probability: A Graduate Course, 2nd edn. Springer, New York.
• Janson, S. (2009). The probability that a random multigraph is simple. Combinatorics Prob. Comput. 18, 205–225.
• Janson, S. and Luczak, M. J. (2008). Asymptotic normality of the $k$-core in random graphs. Ann. Appl. Prob. 18, 1085–1137.
• Janson, S., Luczak, M. and Windridge, P. (2014). Law of large numbers for the SIR epidemic on a random graph with given degrees. Random Structures Algorithms 45, 724–761.
• McKay, B. D. (1984). Asymptotics for $0$-$1$ matrices with prescribed line sums. In Enumeration and Design (Waterloo, Ontario, 1982), Academic Press, Toronto, ON, pp. 225–238.
• McKay, B. D. (1985). Asymptotics for symmetric $0$-$1$ matrices with prescribed row sums. Ars Combinatoria 19A, 15–25.
• McKay, B. D. and Wormald, N. C. (1991). Asymptotic enumeration by degree sequence of graphs with degrees $o(n\sp {1/2})$. Combinatorica 11, 369–382.
• Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). NIST Handbook of Mathematical Functions. Cambridge University Press.
• Wormald, N. C. (1978). Some problems in the enumeration of labelled graphs. Doctoral Thesis, University of Newcastle.
• Wormald, N. C. (1981). The asymptotic distribution of short cycles in random regular graphs. J. Combinatorial Theory B 31, 168–182.