Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-12-03T19:15:33.631Z Has data issue: false hasContentIssue false
  • English
  • Français

The Final Size of the C4-Free Process

Published online by Cambridge University Press:  05 September 2011

MICHAEL E. PICOLLELLI
MICHAEL E. PICOLLELLI*
Affiliation:
Department of Electrical and Computer Engineering, University of Delaware, Newark, DE, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C4. We show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O((nlogn)1/3) and consequently size O(n4/3(logn)1/3), which are sharp up to constants. This confirms conjectures of Bohman and Keevash and of Osthus and Taraz, and improves upon previous bounds due to Bollobás and Riordan and Osthus and Taraz.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 20 , Issue 6 , November 2011 , pp. 939 - 955
Copyright
Copyright © Cambridge University Press 2011

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]Alon, N., Krivelevich, M. and Sudakov, B. (1999) Coloring graphs with sparse neighborhoods. J. Combin. Theory Ser. B 77 73–82.CrossRefGoogle Scholar
[2]Bohman, T. (2009) The triangle-free process. Adv. Math. 221 1653–1677.CrossRefGoogle Scholar
[3]Bohman, T. and Keevash, P. (2010) The early evolution of the H-free process. Invent. Math. 181 291–336.CrossRefGoogle Scholar
[4]Bollobás, B. (2001) Random Graphs, second edition, Cambridge University Press.CrossRefGoogle Scholar
[5]Bollobás, B. and Riordan, O. (2000) Constrained graph processes. Electron. J. Combin. 7 #R18.CrossRefGoogle Scholar
[6]Erdős, P., Suen, S. and Winkler, P. (1995) On the size of a random maximal graph. Random Struct. Alg. 6 309–318.CrossRefGoogle Scholar
[7]Gerke, S. and Makai, T. (2010) No dense subgraphs appear in the triangle-free graph process. Manuscript. arXiv:1002.2316Google Scholar
[8]Kim, J. H. (1995) The Ramsey number R(3, t) has order of magnitude t 2/logt. Random Struct. Alg. 7 173–207.CrossRefGoogle Scholar
[9]Osthus, D. and Taraz, A. (2001) Random maximal H-free graphs. Random Struct. Alg. 18 61–82.3.0.CO;2-T>CrossRefGoogle Scholar
[10]Picollelli, M. (2010) The diamond-free process. Manuscript. arXiv:1010.5207Google Scholar
[11]Ruciński, A. and Wormald, N. (1992) Random graph processes with degree restrictions. Combin. Probab. Comput. 1 169–180.CrossRefGoogle Scholar
[12]Spencer, J. (1990) Counting extensions. J. Combin. Theory Ser. A 55 247–255.CrossRefGoogle Scholar
[13]Spencer, J. Maximal trianglefree graphs and Ramsey R(3, k). Unpublished manuscript. www.cs.nyu.edu/spencer/papers/ramsey3k.pdf.Google Scholar
[14]Warnke, L. Dense subgraphs in the H-free process. Discrete Math., to appear. arXiv:1003.0220Google Scholar
[15]Warnke, L. (2010) When does the K 4-free process stop? Manuscript. arXiv:1007.3037Google Scholar
[16]Wolfovitz, G. (2009) Lower bounds for the size of maximal H-free graphs. Electron. J. Combin. 16 #R4.CrossRefGoogle Scholar
[17]Wolfovitz, G. (2010) The K 4-free process. Manuscript. arXiv:1008.4044Google Scholar
[18]Wolfovitz, G. Triangle-free subgraphs in the triangle-free process. Random Struct. Alg., to appear. arXiv:0903.1756Google Scholar
[19]Wormald, N. (1999) The differential equation method for random graph processes and greedy algorithms. In Lectures on Approximation and Randomized Algorithms (Karonski, M. and Prömel, H. J., eds), pp. 73–155, PWN, Warsaw.Google Scholar