A Sharp Dirac–Erdős Type Bound for Large Graphs

Let k ⩾ 3 be an integer, h_k (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 h_k (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 G_k (n) with h_k (G_k (n)) − ℓ_k(G_k (n)) = 2k − 1 such that G_k (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 h_k (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 h_k (G) − ℓ_k(G) ⩾ 2k but no k disjoint cycles. In particular, every graph G with |V(G)| ⩾ 19k and h_k (G) − ℓ_k(G) ⩾ 2k contains k disjoint cycles.