A GENERALIZATION OF ORE THEOREM INVOLVING NEIGHBORHOOD UNIONS

Let G be a graph of order n. Settling conjectures of Chen and Jackson, we prove the following generalization of Ore's Theorem: If G is 2-con nected and \N(u) or N(v)\ greater-than-or-equal-to 1/2n for every pair of nonadjacent vertices u, v, then either G is hamiltonian, or G is t he Petersen graph, or G belongs to one of three families of exceptiona l graphs of connectivity 2.