Improved degree conditions for Hamiltonian properties

In 1980, Bondy proved that for an integer k ≥ 2 a (k+ s)-connected graph of order n ≥ 3 is traceable (s = −1) or Hamiltonian (s = 0) or Hamiltonian-connected (s = 1) if the degree sum of every set of k+1 pairwise nonadjacent vertices is at least 1 2 ((k+1)(n+ s−1)+1). This generalizes the well-known sufficient conditions of Dirac (k = 0) and Ore (k = 1). The condition in Bondy’s Theorem is not tight for k ≥ 2. We improve this sufficient degree condition and show the general tightness of this result. © 2011 Elsevier B.V. All rights reserved.