Pósa proved that a random graph with cn log n edges is Hamiltonian with probability tending to 1 if c > 3. Korsunov improved this by showing that, if Gn is a random graph with frac(1, 2) n log n + frac(1, 2) n log log n + f (n) n edges and f (n) → ∞, then Gn is Hamiltonian, with probability tending to 1. We shall prove that if a graph Gn has n vertices and frac(1, 2) n log n + frac(1, 2) n log log n + cn edges, then it is Hamiltonian with probability Pc tending to exp exp (- 2 c) as n → ∞. © 1983.
Komlós, J., & Szemerédi, E. (2006). Limit distribution for the existence of Hamiltonian cycles in a random graph. Discrete Mathematics, 306(10–11), 1032–1038. https://doi.org/10.1016/j.disc.2006.03.022