### Abstract

Original language | English |
---|---|

Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 10 |

Publication status | Published - 2001 |

### Publication series

Name | Memorandum |
---|---|

Publisher | Department of Applied Mathematics, University of Twente |

No. | 1576 |

ISSN (Print) | 0169-2690 |

### Fingerprint

### Keywords

- MSC-05C45
- IR-65763
- EWI-3396
- MSC-05C35

### Cite this

*Toughness and hamiltonicity in $k$-trees*. (Memorandum; No. 1576). Enschede: University of Twente, Department of Applied Mathematics.

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*Toughness and hamiltonicity in $k$-trees*. Memorandum, no. 1576, University of Twente, Department of Applied Mathematics, Enschede.

**Toughness and hamiltonicity in $k$-trees.** / Broersma, H.J.; Xiong, L.; Yoshimoto, K.

Research output: Book/Report › Report › Other research output

TY - BOOK

T1 - Toughness and hamiltonicity in $k$-trees

AU - Broersma, H.J.

AU - Xiong, L.

AU - Yoshimoto, K.

N1 - Imported from MEMORANDA

PY - 2001

Y1 - 2001

N2 - We consider toughness conditions that guarantee the existence of a hamiltonian cycle in $k$-trees, a subclass of the class of chordal graphs. By a result of Chen et al.\ 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al.\ there exist nontraceable chordal graphs with toughness arbitrarily close to $\frac{7}{4}$. It is believed that the best possible value of the toughness guaranteeing hamiltonicity of chordal graphs is less than 18, but the proof of Chen et al.\ indicates that proving a better result could be very complicated. We show that every 1-tough 2-tree on at least three vertices is hamiltonian, a best possible result since 1-toughness is a necessary condition for hamiltonicity. We generalize the result to $k$-trees for $k\ge 2$: Let $G$ be a $k$-tree. If $G$ has toughness at least $\frac{k+1}{3},$ then $G$ is hamiltonian. Moreover, we present infinite classes of nonhamiltonian 1-tough $k$-trees for each $k\ge 3$

AB - We consider toughness conditions that guarantee the existence of a hamiltonian cycle in $k$-trees, a subclass of the class of chordal graphs. By a result of Chen et al.\ 18-tough chordal graphs are hamiltonian, and by a result of Bauer et al.\ there exist nontraceable chordal graphs with toughness arbitrarily close to $\frac{7}{4}$. It is believed that the best possible value of the toughness guaranteeing hamiltonicity of chordal graphs is less than 18, but the proof of Chen et al.\ indicates that proving a better result could be very complicated. We show that every 1-tough 2-tree on at least three vertices is hamiltonian, a best possible result since 1-toughness is a necessary condition for hamiltonicity. We generalize the result to $k$-trees for $k\ge 2$: Let $G$ be a $k$-tree. If $G$ has toughness at least $\frac{k+1}{3},$ then $G$ is hamiltonian. Moreover, we present infinite classes of nonhamiltonian 1-tough $k$-trees for each $k\ge 3$

KW - MSC-05C45

KW - IR-65763

KW - EWI-3396

KW - MSC-05C35

M3 - Report

T3 - Memorandum

BT - Toughness and hamiltonicity in $k$-trees

PB - University of Twente, Department of Applied Mathematics

CY - Enschede

ER -