Fan-type condition on disjoint cycles in a graph

Let k be a positive integer. In this paper, we prove that for a graph G with at least 4k vertices, if max{d(x),d(y)}≥2k for any pair of nonadjacent vertices {x,y}⊆V(G), then G contains k disjoint cycles. This generalizes the results given by Corrá di and Hajnal (1963), Enomoto (1998), and Wang (1999).