## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1-11 |

Number of pages | 11 |

Journal | Combinatorics Probability and Computing |

DOIs | |

State | Accepted/In press - Mar 9 2018 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics