Random trees and random graphs
proceedings of the eighth international conference on Random structures and algorithms
Regular Article: The Diameter of Sparse Random Graphs
Advances in Applied Mathematics
The phase transition in inhomogeneous random graphs
Random Structures & Algorithms
The diameter of sparse random graphs
Random Structures & Algorithms
Anatomy of a young giant component in the random graph
Random Structures & Algorithms
The diameter of sparse random graphs
Combinatorics, Probability and Computing
The evolution of the cover time
Combinatorics, Probability and Computing
Anatomy of the giant component: The strictly supercritical regime
European Journal of Combinatorics
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We study the diameter of 1, the largest component of the Erdős–Rényi random graph (n, p) in the emerging supercritical phase, i.e., for p = $\frac{1+\epsilon}n$ 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. Prior to this work, Riordan and Wormald gave precise estimates on the diameter; however, these did not cover the entire supercritical regime (namely, when ε3n → ∞ arbitrarily slowly). Łuczak and Seierstad estimated its order throughout this regime, yet their upper and lower bounds differed by a factor of $\frac{1000}7$. We show that throughout the emerging supercritical phase, i.e., for any ε = o(1) with ε3n → ∞, the diameter of 1 is with high probability asymptotic to D(ε, n) = (3/ε)log(ε3n). This constitutes the first proof of the asymptotics of the diameter valid throughout this phase. The proof relies on a recent structure result for the supercritical giant component, which reduces the problem of estimating distances between its vertices to the study of passage times in first-passage percolation. The main advantage of our method is its flexibility. It also implies that in the emerging supercritical phase the diameter of the 2-core of 1 is w.h.p. asymptotic to $\frac23 D(\epsilon,n)$, and the maximal distance in 1 between any pair of kernel vertices is w.h.p. asymptotic to $\frac{5}9D(\epsilon,n)$.