Hamiltonicity, independence number, and pancyclicity

A graph on n vertices is called pancyclic if it contains a cycle of length ℓ for all 3 ≤ ℓ ≤ n . In 1972, Erdős proved that if G is a Hamiltonian graph on n > 4 k 4 vertices with independence number k , then G is pancyclic. He then suggested that n = Ω ( k 2 ) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such that n > c k 7 / 3 suffices.