## Electronic Communications in Probability

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

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

#### Abstract

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

**Source**

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

**Dates**

Received: 4 May 2016

Accepted: 30 August 2016

First available in Project Euclid: 14 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1473854580

**Digital Object Identifier**

doi:10.1214/16-ECP20

**Zentralblatt MATH identifier**

1346.60006

**Subjects**

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

**Keywords**

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

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

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. https://projecteuclid.org/euclid.ecp/1473854580