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.