@book{980a12586fd548deb8cece8ce27662af,
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, MSC-05C35",
author = "H.J. Broersma and L. Xiong and K. Yoshimoto",
year = "2001",
language = "English",
series = "Memorandum",
publisher = "University of Twente",
number = "1576",
address = "Netherlands",
}