### 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) |
---|---|

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 |
---|---|

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

**An algorithmic answer to the ore-type version of Dirac’s question on disjoint cycles.** / Kierstead, Henry; Kostochka, A. V.; Molla, T.; Yager, D.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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

}

TY - CHAP

T1 - An algorithmic answer to the ore-type version of Dirac’s question on disjoint cycles

AU - Kierstead, Henry

AU - Kostochka, A. V.

AU - Molla, T.

AU - Yager, D.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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).

AB - 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).

UR - http://www.scopus.com/inward/record.url?scp=85054178766&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85054178766&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-94830-0_8

DO - 10.1007/978-3-319-94830-0_8

M3 - Chapter

AN - SCOPUS:85054178766

T3 - Springer Optimization and Its Applications

SP - 149

EP - 168

BT - Springer Optimization and Its Applications

PB - Springer International Publishing

ER -