A Sharp Dirac-Erdos 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 Erdos proved in 1963 that if h_k(G) - ℓ_k(G) ≥ k² + 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)) - ℓk(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 G₀(k) with h_k(G₀(k)) - ℓk(G₀(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-Erdos 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.