On the Set of Cycle Lengths in a Hamiltonian Graph with a Given Maximum Degree

Abstract.

Let n and Δ be two integers such that 2≤Δ≤n−1. We describe the set of cycle lengths occurring in any hamiltonian graph G of order n and maximum degree Δ. We conclude that for the case this set contains all the integers belonging to the union [3,2Δ−n+2]∪[n−Δ+2,Δ+1], and for it contains every integer between 3 and Δ+1. We also study the set of cycle lengths in a hamiltonian graph with two fixed vertices of large degree sum. Our main results imply that the stability s(P) for the property of being pancyclic satisfies