Limit distribution for the existence of Hamiltonian cycles in a random graph

AbstractPósa proved that a random graph with cnlogn edges is Hamiltonian with probability tending to 1 if c>3. Korsunov improved this by showing that, if Gn is a random graph with 12nlogn+12nloglogn+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 12nlogn+12nloglogn+cn edges, then it is Hamiltonian with probability Pc tending to expexp(-2c) as n→∞