Improved degree conditions for Hamiltonian properties

In 1980, Bondy proved that for an integer k≥2k≥2 a (k+s)(k+s)-connected graph of order n≥3n≥3 is traceable (s=−1s=−1) or Hamiltonian (s=0s=0) or Hamiltonian-connected (s=1s=1) if the degree sum of every set of k+1k+1 pairwise nonadjacent vertices is at least 12((k+1)(n+s−1)+1). This generalizes the well-known sufficient conditions of Dirac (k=0k=0) and Ore (k=1k=1). The condition in Bondy’s Theorem is not tight for k≥2k≥2. We improve this sufficient degree condition and show the general tightness of this result.