Let k ⩾ 3 be an integer, hk (G) be the number of vertices of degree at least 2k in a graph G, and ℓ k (G) be the number of vertices of degree at most 2k − 2 in G. Dirac and Erdős proved in 1963 that if hk (G) − ℓ k (G) ⩾ k 2 + 2k − 4, then G contains k vertex-disjoint cycles. For each k ⩾ 2, they also showed an infinite sequence of graphs Gk (n) with hk (Gk (n)) − ℓ k (Gk (n)) = 2k − 1 such that Gk (n) does not have k disjoint cycles. Recently, the authors proved that, for k ⩾ 2, a bound of 3k is sufficient to guarantee the existence of k disjoint cycles, and presented for every k a graph G 0(k) with hk (G 0(k)) − ℓ k (G 0(k)) = 3k − 1 and no k disjoint cycles. The goal of this paper is to refine and sharpen this result. We show that the Dirac–Erdős construction is optimal in the sense that for every k ⩾ 2, there are only finitely many graphs G with hk (G) − ℓ k (G) ⩾ 2k but no k disjoint cycles. In particular, every graph G with |V(G)| ⩾ 19k and hk (G) − ℓ k (G) ⩾ 2k contains k disjoint cycles.
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