Pancyclism and small cycles in graphs

We first show that if a graph G of order n contains a hamiltonian path connecting two nonadjacent vertices u and v such that d(u) + d(v) ≥ n, then G is pancyclic. By using this result, we prove that if G is hamiltonian with order n ≥ 20 and if G has two nonadjacent vertices u and v such that d(u) + d(v) ≥ n + z, where z = 0 when n is odd and z = 1 otherwise, then G contains a cycle of length m for each 3 ≤ m ≤ max (dC(u, v) + 1, n+19 13 ), dC(u, v) being the distance of u and v on a hamiltonian cycle of G.