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Diameters in supercritical random graphs via first passage percolation

By Jian Ding, Jeong Han Kim, Eyal Lubetzky and Yuval Peres

Abstract

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

Topics: D(ε, n
Year: 2011
OAI identifier: oai:CiteSeerX.psu:10.1.1.188.574
Provided by: CiteSeerX
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