Cycle Lengths of Hamiltonian Pℓ-free Graphs

For an integer ℓ at least three, we prove that every Hamiltonian P ℓ -free graph G on n > ℓ vertices has cycles of at least 2 ℓ n - 1 different lengths. For small values of ℓ , we can improve the bound as follows. If 4 ≤ ℓ ≤ 7 , then G has cycles of at least 1 2 n - 1 different lengths, and if ℓ is 4 or 5 and n is odd, then G has cycles of at least n - ℓ + 2 different lengths.