Pancyclicity of Hamiltonian graphs

An n -vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from 3 up to n. In 1972, Erdős conjectured that every Hamiltonian graph with independence number at most k and at least n = (k^2) vertices is pancyclic. We prove this old conjecture in a strong form by showing that if such a graph has n = (2+o (1) ) k^2 vertices, it is already pancyclic, and this bound is asymptotically best possible.