Hostname: page-component-74656f9884-rdgjk Total loading time: 0 Render date: 2026-11-07T11:20:19.155Z Has data issue: false hasContentIssue false
  • English
  • Français

Subgraphs of Dense Random Graphs with Specified Degrees

Published online by Cambridge University Press:  27 January 2011

BRENDAN D. McKAY
BRENDAN D. McKAY*
Affiliation:
School of Computer Science, Australian National University, Canberra ACT 0200, Australia (e-mail: bdm@cs.anu.edu.au)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Let d = (d1, d2, . . ., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.

Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n.

Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory developed for bipartite graphs by Greenhill and McKay (2009).

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 3 , May 2011 , pp. 413 - 433
Copyright
Copyright © Cambridge University Press 2011

References

[1]Barvinok, A. and Hartigan, J. A. (2010) The number of graphs and a random graph with a given degree sequence. Preprint, available at arxiv.org/abs/1003.0356.Google Scholar
[2]Boldi, P. and Vigna, S. (2003) Lower bounds for sense of direction in regular graphs. Distrib. Comput. 16 279–286.CrossRefGoogle Scholar
[3]Bollobás, B. and McKay, B. D. (1986) The number of matchings in random regular graphs and bipartite graphs. J. Combin. Theory Ser. B 41 80–91.CrossRefGoogle Scholar
[4]Canfield, E. R., Greenhill, C. and McKay, B. D. (2008) Asymptotic enumeration of dense 0–1 matrices with specified line sums. J. Combin. Theory Ser. A 115 32–66.Google Scholar
[5]Chatterjee, S., Diaconis, P. and Sly, A. (2010) Random graphs with a given degree sequence. Preprint, available at arxiv.org/abs/1005.1136.Google Scholar
[6]Cooper, C., Frieze, A. and Reed, B. (2002) Random regular graphs of non-constant degree: Connectivity and Hamiltonicity. Combin. Probab. Comput. 11 249–261.CrossRefGoogle Scholar
[7]Cooper, C., Frieze, A., Reed, B. and Riordan, O. (2002) Random regular graphs of non-constant degree: Independence and chromatic number. Combin. Probab. Comput. 11 323–341.CrossRefGoogle Scholar
[8]Greenhill, C. and McKay, B. D. (2009) Random dense bipartite graphs and directed graphs with specified degrees. Random Struct. Alg. 35 222–249.CrossRefGoogle Scholar
[9]Greenhill, C., McKay, B. D. and Wang, X. (2006) Asymptotic enumeration of sparse irregular bipartite graphs. J. Combin. Theory Ser. A 113 291–324.CrossRefGoogle Scholar
[10]Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.CrossRefGoogle Scholar
[11]Krivelevich, M., Sudakov, B. and Wormald, N. C. (2010) Regular induced subgraphs of a random graph. Random Struct. Alg., published online, doi: 10.1002/rsa.20324.CrossRefGoogle Scholar
[12]Krivelevich, M., Sudakov, B., Vu, V. and Wormald, N. C. (2001) Random regular graphs of high degree. Europ. J. Combin. 18 346–363.Google Scholar
[13]McKay, B. D. (1981) Spanning trees in random regular graphs. In Third Caribbean Conference on Combinatorics and Computing (University of West Indies 1981), pp. 139–143.Google Scholar
[14]McKay, B. D. (1981) Subgraphs of random graphs with specified degrees. Congr. Numer. 33 213–223.Google Scholar
[15]McKay, B. D. (1985) Asymptotics for symmetric 0–1 matrices with prescribed row sums. Ars Combin. 19A 15–26.Google Scholar
[16]McKay, B. D. (1990) The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs. Combinatorica 10 367–377.CrossRefGoogle Scholar
[17]McKay, B. D. and Robinson, R. W. (1998) Asymptotic enumeration of Eulerian circuits in the complete graph. Combin. Probab. Comput. 7 437–449.CrossRefGoogle Scholar
[18]McKay, B. D. and Wang, X. (1996) Asymptotic enumeration of tournaments with a given score sequence. J. Combin. Theory Ser. A 73 77–90.CrossRefGoogle Scholar
[19]McKay, B. D. and Wormald, N. C. (1990) Asymptotic enumeration by degree sequence of graphs of high degree. European J. Combin. 11 565–580.CrossRefGoogle Scholar
[20]McKay, B. D. and Wormald, N. C. (1991) Asymptotic enumeration by degree sequence of graphs with degrees o(n 1/2). Combinatorica 11 369–382.CrossRefGoogle Scholar
[21]Wormald, N. C. (1999) Models of random regular graphs. In Surveys in Combinatorics 1999 (Lamb, J. D. and Preece, D. A., eds), Cambridge University Press, pp. 239–298.CrossRefGoogle Scholar