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].