On pancyclism of hamiltonian graphs

Let n and Δ be two integers with Δ ≤ n − 1. We study the set of cycle lengths occurring in any hamiltonian graph G of order n and maximum degree Δ. We show that for the case n/2+1 ≤ Δ ≤ 2n-2/3 this set contains all the integers belonging to the union [3, 2Δ-n+2] ∪ [n-Δ+ 2,Δ+1], and for 2n−23 ≤ Δ ≤ n − 1 it contains every integer between 3 and Δ + 1. We also investigate the set of cycle lengths in a hamiltonian graph with two fixed vertices of large degree sum.