Chordality and 2-factors in tough graphs

Bauer, Douglas C.; Katona, Gyula O. H.; Kratsch, Dieter; Veldman, H.J.

doi:10.1016/s0166-218x(99)00142-0

Cited by 17 publications

(42 citation statements)

References 8 publications

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“…On the other hand, a construction of (7/4 − ε)-tough non-hamiltonian chordal graphs was given in [1]. It was conjectured in [2] that every 2-tough chordal graph is hamiltonian. Let us notice at this point that chordal graphs form a superclass of both interval graphs and split graphs.…”

Section: Conjecture 1 There Exists a Constant β Such That Every β-Tomentioning

confidence: 99%

Tough spiders

Kaiser

Kráľ²,

Stacho

2007

Journal of Graph Theory

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Section: Conjecture 1 There Exists a Constant β Such That Every β-Tomentioning

confidence: 99%

Tough spiders

Kaiser

Kráľ²,

Stacho

2007

Journal of Graph Theory

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“…These examples are all chordal. Recently it was shown in [4] that every 3 2 -tough chordal graph has a 2-factor. Based on this, Kratsch [14] raised the question whether every 3 2 -tough chordal graph is hamiltonian.…”

Section: Special Graph Classesmentioning

confidence: 99%

Toughness and hamiltonicity in k-trees

Broersma¹,

Xiong

Yoshimoto

2007

Discrete Mathematics

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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 1-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 1-tough k-trees for each k 3.

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“…These examples are all chordal. Recently, it was shown by Bauer et al [4] that every 3 2 -tough chordal graph has a 2-factor. Based on this, Kratsch [51] raised the question whether every 3 2 -tough chordal graph is hamiltonian.…”

Section: Chordal Graphsmentioning

confidence: 99%