Let k ≥ 3 be an integer, hk(G) be the number of vertices of degree at least 2k in a graph G, and ℓk(G) be the number of vertices of degree at most 2k - 2 in G. Dirac and Erdos proved in 1963 that if hk(G) - ℓk(G) ≥ k2 + 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)) - ℓk(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 G0(k) with hk(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 hk(G) - ℓk(G) ≥ 2k but no k disjoint cycles. In particular, every graph G with |V(G)| ≥ 19k and hk(G) - ℓk(G) ≥ 2k contains k disjoint cycles.
ASJC Scopus subject areas
- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics