Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T03:46:09.729Z Has data issue: false hasContentIssue false
  • English
  • Français

Counting Connected Hypergraphs via the Probabilistic Method

Published online by Cambridge University Press:  04 January 2016

BÉLA BOLLOBÁS  and
OLIVER RIORDAN
BÉLA BOLLOBÁS
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB, UK and Department of Mathematical Sciences, University of Memphis, Memphis TN 38152, USA (e-mail: [email protected])
OLIVER RIORDAN
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on [n] = {1,2,. . .,n} with m edges, whenever n and the nullity m−n+1 tend to infinity. Let Cr(n,t) be the number of connected r-uniform hypergraphs on [n] with nullity t = (r−1)m−n+1, where m is the number of edges. For r ≥ 3, asymptotic formulae for Cr(n,t) are known only for partial ranges of the parameters: in 1997 Karoński and Łuczak gave one for t = o(log n/log log n), and recently Behrisch, Coja-Oghlan and Kang gave one for t=Θ(n). Here we prove such a formula for any fixed r ≥ 3 and any t = t(n) satisfying t = o(n) and t→∞ as n→∞, complementing the last result. This leaves open only the case t/n→∞, which we expect to be much simpler, and will consider in future work. The proof is based on probabilistic methods, and in particular on a bivariate local limit theorem for the number of vertices and edges in the largest component of a certain random hypergraph. We deduce this from the corresponding central limit theorem by smoothing techniques.

Keywords

Primary 05C80Secondary 05C6505C30
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 1: Oberwolfach Special Issue Part 2 , January 2016 , pp. 21 - 75
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Andriamampianina, T. and Ravelomanana, V. (2005) Enumeration of connected uniform hypergraphs. In Proc. 17th International Conference on Formal Power Series and Algebraic Combinatorics, Taormina: FPSAC 2005, pp. 387–398.Google Scholar
[2]Behrisch, M., Coja-Oghlan, A. and Kang, M. (2007) Local limit theorems and number of connected hypergraphs. arXiv:0706.0497Google Scholar
[3]Behrisch, M., Coja-Oghlan, A. and Kang, M. (2007) Local limit theorems for the giant component of random hypergraphs. In Proc. RANDOM 2007, Vol. 4627 of Lecture Notes in Computer Science, Springer, pp. 341–352.Google Scholar
[4]Behrisch, M., Coja-Oghlan, A. and Kang, M. (2010) The order of the giant component of random hypergraphs. Random Struct. Alg. 36 149–184.CrossRefGoogle Scholar
[5]Behrisch, M., Coja-Oghlan, A. and Kang, M. (2014) Local limit theorems for the giant component of random hypergraphs. Combin. Probab. Comput. 23 331–366.CrossRefGoogle Scholar
[6]Behrisch, M., Coja-Oghlan, A. and Kang, M. (2014) The asymptotic number of connected d-uniform hypergraphs. Combin. Probab. Comput. 23 367–385. Corrigendum: Combin. Probab. Comput. 24 (2015) 373–375.CrossRefGoogle Scholar
[7]Bender, E. A., Canfield, E. R. and McKay, B. D. (1990) The asymptotic number of labeled connected graphs with a given number of vertices and edges. Random Struct. Alg. 1 127–169.CrossRefGoogle Scholar
[8]Bollobás, B. (1984) The evolution of sparse graphs. In Graph Theory and Combinatorics: Cambridge 1983, Academic Press, pp. 35–57.Google Scholar
[9]Bollobás, B. and Riordan, O. (2012) Asymptotic normality of the size of the giant component in a random hypergraph. Random Struct. Alg. 41 441–450.CrossRefGoogle Scholar
[10]Bollobás, B. and Riordan, O. Exploring hypergraphs with martingales. Random Struct. Alg., to appear. arXiv:1403.6558Google Scholar
[11]Bollobás, B. and Riordan, O. Counting connected hypergraphs via the probabilistic method, with appendix. arXiv:1404.5887Google Scholar
[12]Bollobás, B. and Riordan, O. Counting dense connected hypergraphs via the probabilistic method. arXiv:1511.04739Google Scholar
[13]Cayley, A. (1889) A theorem on trees. Quart. J. Pure and Applied Math. 23 376–378.Google Scholar
[14]Coja-Oghlan, A., Moore, C. and Sanwalani, V. (2007) Counting connected graphs and hypergraphs via the probabilistic method. Random Struct. Alg. 31 288–329.CrossRefGoogle Scholar
[15]Davis, B. and McDonald, D. (1995) An elementary proof of the local central limit theorem. J. Theoret. Probab. 8 693–701.CrossRefGoogle Scholar
[16]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.Google Scholar
[17]Karoński, M. and Łuczak, T. (1997) The number of connected sparsely edged uniform hypergraphs. Discrete Math. 171 153–167.CrossRefGoogle Scholar
[18]Karoński, M. and Łuczak, T. (2002) The phase transition in a random hypergraph. J. Comput. Appl. Math. 142 125–135.CrossRefGoogle Scholar
[19]Luczak, M. and Łuczak, T. (2006) The phase transition in the cluster-scaled model of a random graph. Random Struct. Alg. 28 215–246.CrossRefGoogle Scholar
[20]McDonald, D. R. (1979) On local limit theorem for integer valued random variables. Teor. Veroyatnost. i Primenen. 24 607–614. See also Theory Probab. Appl. 24 (1980) 613–619.Google Scholar
[21]Pittel, B. and Wormald, C. (2005) Counting connected graphs inside-out. J. Combin. Theory Ser. B 93 127–172.CrossRefGoogle Scholar
[22]Rényi, A. (1959) On connected graphs I (Hungarian, Russian summary). Magyar Tud. Akad. Mat. Kutató Int. Közl. 4 385–388.Google Scholar
[23]Sato, C. M. (2013) Core structures in random graphs and hypergraphs. PhD thesis, Department of Combinatorics and Optimization, University of Waterloo. https://uwspace.uwaterloo.ca/handle/10012/7787Google Scholar
[24]Sato, C. M. and Wormald, N. Asymptotic enumeration of sparse connected 3-uniform hypergraphs. arXiv:1401.7381Google Scholar
[25]Schmidt-Pruzan, J. and Shamir, E. (1985) Component structure in the evolution of random hypergraphs. Combinatorica 5 81–94.CrossRefGoogle Scholar
[26]Scott, A. and Tateno, A. On the number of triangles in a random graph. Manuscript.Google Scholar
[27]Selivanov, B. I. (1972) Enumeration of homogeneous hypergraphs with a simple cycle structure. Kombinatornyĭ Anal. 2 60–67.Google Scholar
[28]Wright, E. M. (1977) The number of connected sparsely edged graphs. J. Graph Theory 1 317–330.CrossRefGoogle Scholar
[29]Wright, E. M. (1978) The number of connected sparsely edged graphs II: Smooth graphs and blocks. J. Graph Theory 2 299–305.CrossRefGoogle Scholar
[30]Wright, E. M. (1980) The number of connected sparsely edged graphs III: Asymptotic results. J. Graph Theory 4 393–407.CrossRefGoogle Scholar
[31]Wright, E. M. (1983) The number of connected sparsely edged graphs {IV}: Large nonseparable graphs. J. Graph Theory 7 219–229.CrossRefGoogle Scholar