# 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

• 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.