Disjoint cycles in graphs with distance degree sum conditions

In this paper, we prove that for any positive integer k and for any simple graph G of order at least 3k, if d(x)+d(y)≥4k for every pair of vertices x and y of distance 2 in G, then with one exception, G contains k disjoint cycles. This generalizes a former result of Corrádi and Hajnal (1963).