### Abstract

A graph G is said to be n-factor-critical if G - S has a 1-factor for any S ⊂ V(G) with |S| = n. In this paper, we prove that if G is a 2-connected n-factor-critical graph of order p with σ_{3}3/2(G)≥(p - n - 1), then G is hamiltonian with some exceptions. To extend this theorem, we define a (k,n)-factor-critical graph to be a graph G such that G - S has a k-factor for any S ⊂ V(G) with |S| = n. We conjecture that if G is a 2-connected (k,n)-factor-critical graph of order p with σ_{3}(G)≥3/2(p - n - k), then G is hamiltonian with some exceptions. In this paper, we characterize all such graphs that satisfy the assumption, but are not 1-tough. Using this, we verify the conjecture for k≤2.

Original language | English |
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Pages (from-to) | 71-82 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 240 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 2001 Sep 28 |

### Keywords

- Degree sum
- Factor-critical graphs
- Hamiltonian cycle
- Toughness

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

*Discrete Mathematics*,

*240*(1-3), 71-82. https://doi.org/10.1016/S0012-365X(00)00386-1