An improved lower bound on the length of the longest cycle in random graphs

08/14/2022

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by   Michael Anastos, et al.

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We provide a new lower bound on the length of the longest cycle of the binomial random graph G(n,(1+ϵ)/n) that holds w.h.p. for all ϵ=ϵ(n) such that ϵ^3n→∞. In the case ϵ≤ϵ_0 for some sufficiently small constant ϵ_0, this bound is equal to 1.581ϵ^2n which improves upon the current best lower bound of 4ϵ^2n/3 due to Luczak.