### Abstract

Corrádi and Hajnal in 1963 proved the following theorem on the NP-complete problem on the existence of k disjoint cycles in an n-vertex graph G: For all k ≥ 1 and n ≥ 3k, every (simple) n-vertex graph G with minimum degree δ(G) ≥ 2k contains k disjoint cycles. The same year, Dirac described the 3-connected multigraphs not containing two disjoint cycles and asked the more general question: Which (2k − 1)-connected multigraphs do not contain k disjoint cycles? Recently, Kierstead, Kostochka, and Yeager resolved this question. In this paper, we sharpen this result by presenting a description that can be checked in polynomial time of all multigraphs G with no k disjoint cycles for which the underlying simple graph G̲ satisfies the following Ore-type condition: dG̲(v)+dG̲(u)≥4k−3 for all nonadjacent u, v ∈ V (G).

Original language | English (US) |
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Title of host publication | Springer Optimization and Its Applications |

Publisher | Springer International Publishing |

Pages | 149-168 |

Number of pages | 20 |

DOIs | |

State | Published - Jan 1 2018 |

### Publication series

Name | Springer Optimization and Its Applications |
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Volume | 139 |

ISSN (Print) | 1931-6828 |

ISSN (Electronic) | 1931-6836 |

### Fingerprint

### ASJC Scopus subject areas

- Control and Optimization

### Cite this

*Springer Optimization and Its Applications*(pp. 149-168). (Springer Optimization and Its Applications; Vol. 139). Springer International Publishing. https://doi.org/10.1007/978-3-319-94830-0_8