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)≥n2) 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.