# Toughness and hamiltonicity in $k$-trees

H.J. Broersma, L. Xiong, K. Yoshimoto

Research output: Book/ReportReportOther research output

### Abstract

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 $k\ge 3$
Original language English Enschede University of Twente, Department of Applied Mathematics 10 Published - 2001

### Publication series

Name Memorandum Department of Applied Mathematics, University of Twente 1576 0169-2690

### Fingerprint

K-tree
Hamiltonicity
Toughness
Chordal Graphs
Hamiltonian circuit
Necessary Conditions
Generalise

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

### Cite this

Broersma, H. J., Xiong, L., & Yoshimoto, K. (2001). Toughness and hamiltonicity in $k$-trees. (Memorandum; No. 1576). Enschede: University of Twente, Department of Applied Mathematics.
Broersma, H.J. ; Xiong, L. ; Yoshimoto, K. / Toughness and hamiltonicity in $k$-trees. Enschede : University of Twente, Department of Applied Mathematics, 2001. 10 p. (Memorandum; 1576).
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title = "Toughness and hamiltonicity in $k$-trees",
abstract = "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 $k\ge 3$",
keywords = "MSC-05C45, IR-65763, EWI-3396, MSC-05C35",
author = "H.J. Broersma and L. Xiong and K. Yoshimoto",
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year = "2001",
language = "English",
series = "Memorandum",
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Broersma, HJ, Xiong, L & Yoshimoto, K 2001, 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.

Enschede : University of Twente, Department of Applied Mathematics, 2001. 10 p. (Memorandum; No. 1576).

Research output: Book/ReportReportOther 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 -

Broersma HJ, Xiong L, Yoshimoto K. Toughness and hamiltonicity in $k$-trees. Enschede: University of Twente, Department of Applied Mathematics, 2001. 10 p. (Memorandum; 1576).