Electronic Communications in Probability

On the largest component in the subcritical regime of the Bohman-Frieze process

Sanchayan Sen

Full-text: Open access


Kang, Perkins, and Spencer [7] conjectured that the size of the largest component of the Bohman-Frieze process at a fixed time $t$ smaller than $t_c$, the critical time for the process, is $L_1(t)=O(\log n/(t_c-t)^2)$ with high probability. Bhamidi, Budhiraja, and Wang [3] have shown that a bound of the form $L_1(t_n)=O((\log n)^4/(t_c-t_n)^2)$ holds with high probability for $t_n\leq t_c-n^{-\gamma }$ where $\gamma \in (0,1/4)$. In this paper, we improve the result in [3] by showing that for any fixed $\lambda >0$, $L_1(t_n)=O(\log n/(t_c-t_n)^2)$ with high probability for $t_n\leq t_c-\lambda n^{-1/3}$. In particular, this settles the conjecture in [7].

Article information

Electron. Commun. Probab. Volume 21 (2016), paper no. 64, 15 pp.

Received: 4 May 2016
Accepted: 30 August 2016
First available in Project Euclid: 14 September 2016

Permanent link to this document

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20]

Bohman-Frieze process Achlioptas process bounded-size rules branching process subcritical regime

Creative Commons Attribution 4.0 International License.


Sen, Sanchayan. On the largest component in the subcritical regime of the Bohman-Frieze process. Electron. Commun. Probab. 21 (2016), paper no. 64, 15 pp. doi:10.1214/16-ECP20. http://projecteuclid.org/euclid.ecp/1473854580.

Export citation


  • [1] Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 no. 2, 812–854.
  • [2] Bhamidi, S., Budhiraja, A., and Wang, X. (2014). Bounded-size rules: The barely subcritical regime. Combinatorics, Probability and Computing. 23 no. 4, 505–538.
  • [3] Bhamidi, S., Budhiraja, A., and Wang, X. (2015). Aggregation models with limited choice and the multiplicative coalescent. Random Struct. Alg. 46 no. 1, 55–116.
  • [4] Bollobás, B., Janson, S., and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Struct. Alg. 31 no. 1, 3–122.
  • [5] de la Fortelle, A. (2006). Yule process sample path asymptotics. Elect. Comm. in Probab. 11, 193–199.
  • [6] Janson, S., Łuczak, T., and Ruciński, A. (2000). Random Graphs. John Wiley & Sons. New York
  • [7] Kang, M., Perkins, W., and Spencer, J. (2012). The Bohman-Frieze process near criticality. Random Struct. Alg. 43 no. 2, 221–250.
  • [8] Spencer, J. and Wormald, N. (2007). Birth control for giants. Combinatorica. 27 no. 5, 587–628.