Let G be a graph, and δ ( G ) and α ( G ) be the minimum degree and the independence number of G , respectively. For a vertex v ∈ V ( G ) , d ( v ) and N ( v ) represent the degree of v and the neighborhood of v in G , respectively. A number of sufficient conditions for a connected simple graph G of order n to be Hamiltonian have been proved. Among them are the well known Dirac condition (1952) ( δ ( G ) ≥ n 2 ) and Ore condition (1960) (for any pair of independent vertices u and v , d ( u ) + d ( v ) ≥ n ). In 1984 Fan generalized these two conditions and proved that if G is a 2-connected graph of order n and max { d ( x ) , d ( y ) } ≥ n / 2 for each pair of nonadjacent vertices x , y with distance 2 in G , then G is Hamiltonian. In 1993, Chen proved that if G is a 2-connected graph of order n , and if max { d ( x ) , d ( y ) } ≥ n / 2 for each pair of nonadjacent vertices x , y with 1 ≤ | N ( x ) ∩ N ( y ) | ≤ α ( G ) − 1 , then G is Hamiltonian. In 1996, Chen, Egawa, Liu and Saito further showed that if G is a k -connected graph of order n , and if max { d ( v ) : v ∈ S } ≥ n / 2 for every independent set S of G with | S | = k which has two distinct vertices x , y ∈ S such that the distance between x and y is 2, then G is Hamiltonian. In this paper, we generalize all the above conditions and prove that if G is a k -connected graph of order n , and if max { d ( v ) : v ∈ S } ≥ n / 2 for every independent set S of G with | S | = k which has two distinct vertices x , y ∈ S satisfying 1 ≤ | N ( x ) ∩ N ( y ) | ≤ α ( G ) − 1 , then G is Hamiltonian.