Cycles in hamiltonian graphs of prescribed maximum degree

Let G be a hamiltonian graph G of order n and maximum degree Delta, and let C(G) denote the set of cycle lengths occurring in G. It is easy to see that \C(G) greater than or equal to Delta - 1. In this paper, we prove that if Delta > n/2, then \C(G)j greater than or equal to (n + Delta - 3)/2. We also show that for every Delta greater than or equal to 2 there is a graph G of order n greater than or equal to 2Delta such that \C(G)\ = Delta - 1, and the lower bound in case Delta > n/2 is best possible. (C) 2003 Elsevier Science B.V. All rights reserved.