Neighborhood unions and Hamiltonian properties

AbstractLet G be a simple graph of order n with connectivity k⩾2, independence number α. We prove that if for each independent set S of cardinality k+1, one of the following condition holds: (1) there exist u≠v in S such that d(u)+d(v)⩾n or |N(u)⌢N(v)|⩾α; (2) for any distinct pair u and v in S, |N(u)⌣N(v)|;⩾n−max{d(x)|xϵS}, then G is Hamiltonian. Many known results on Hamiltonian graphs are corollaries of this resul