We study the diameter of C1, the largest component of the Erdős-Rényi random graph G(n, p) emerging from the critical window, i.e., for p = 1+ε n where ε3n → ∞ and ε = o(1). This parameter was extensively studied for fixed ε> 0, yet results for ε = o(1) outside the critical window were only obtained very recently: Riordan and Wormald gave precise estimates on the diameter, however these do not cover the entire supercritical regime (namely, when ε 3 n → ∞ arbitrarily slowly); ̷Luczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differ by a factor of 1000 7. We show that for any ε = o(1) with ε 3 n → ∞, the diameter of C1 is with high probability asymptotic to D(ε, n) = (3/ε) log(ε 3 n). We also prove that, in this regime, the diameter of the 2-core of C1 is w.h.p. asymptotic to 2 D(ε, n), and the maximal distance in it between any 3 pair of kernel vertices is w.h.p. asymptotic to
To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.