### Abstract

In 1963, Corrádi and Hajnal proved that for all k≥1 and n≥3k, every graph G on n vertices with minimum degree δ(G)≥2k contains k disjoint cycles. The bound δ(G)≥2k is sharp. Here we characterize those graphs with δ(G)≥2k−1 that contain k disjoint cycles. This answers the simple-graph case of Dirac's 1963 question on the characterization of (2k−1)-connected graphs with no k disjoint cycles. Enomoto and Wang refined the Corrádi–Hajnal Theorem, proving the following Ore-type version: For all k≥1 and n≥3k, every graph G on n vertices contains k disjoint cycles, provided that d(x)+d(y)≥4k−1 for all distinct nonadjacent vertices x,y. We refine this further for k≥3 and n≥3k+1: If G is a graph on n vertices such that d(x)+d(y)≥4k−3 for all distinct nonadjacent vertices x,y, then G has k vertex-disjoint cycles if and only if the independence number α(G)≤n−2k and G is not one of two small exceptions in the case k=3. We also show how the case k=2 follows from Lovász’ characterization of multigraphs with no two disjoint cycles.

Original language | English (US) |
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Pages (from-to) | 121-148 |

Number of pages | 28 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 122 |

DOIs | |

State | Published - Jan 1 2017 |

### Keywords

- Disjoint cycles
- Equitable coloring
- Graph packing
- Minimum degree
- Ore-degree

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Journal of Combinatorial Theory. Series B*,

*122*, 121-148. https://doi.org/10.1016/j.jctb.2016.05.007