On the Corrádi-Hajnal theorem and a question of Dirac

Enomoto and Wang refined the Corrádi-Hajnal Theorem, proving the following Ore-type version: For all k≥1 and n≥3k, every graph G on n vertices contains k disjoint cycles, provided that d(x)+d(y)≥4k−1 for all distinct nonadjacent vertices x,y. We refine this further for k≥3 and n≥3k+1: If G is a graph on n vertices such that d(x)+d(y)≥4k−3 for all distinct nonadjacent vertices x,y, then G has k vertex-disjoint cycles if and only if the independence number α(G)≤n−2k and G is not one of two small exceptions in the case k=3. We also show how the case k=2 follows from Lovász' characterization of multigraphs with no two disjoint cycles.