### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

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

**Hamiltonian cycles in n-factor-critical graphs.** / Kawarabayashi, Ken Ichi; Ota, Katsuhiro; Saito, Akira.

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 240, no. 1-3, pp. 71-82. https://doi.org/10.1016/S0012-365X(00)00386-1

}

TY - JOUR

T1 - Hamiltonian cycles in n-factor-critical graphs

AU - Kawarabayashi, Ken Ichi

AU - Ota, Katsuhiro

AU - Saito, Akira

PY - 2001/9/28

Y1 - 2001/9/28

N2 - 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 σ33/2(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 σ3(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.

AB - 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 σ33/2(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 σ3(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.

KW - Degree sum

KW - Factor-critical graphs

KW - Hamiltonian cycle

KW - Toughness

UR - http://www.scopus.com/inward/record.url?scp=0035964595&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035964595&partnerID=8YFLogxK

U2 - 10.1016/S0012-365X(00)00386-1

DO - 10.1016/S0012-365X(00)00386-1

M3 - Article

VL - 240

SP - 71

EP - 82

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1-3

ER -