Toughness and hamiltonicity in k-trees

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 (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 I-toughness is a necessary condition for hamiltonicity. We generalize the result to k-trees for k >= 2; Let G be a k-tree. If G has toughness at least (k + 1)/3, then G is hamiltonian. Moreover, we present infinite classes of nonhamiltonian I-tough k-trees for each k >= 3. (c) 2006 Elsevier B.V. All rights reserved.