Abstract
A graphG has toughnesst(G) ift(G) is the largest numbert such that for any subsetS of the vertices ofG, the number of vertices inS is at leastt times the number of components ofG on deletion of the vertices inS, provided that there is then more than one component. Ak-tree of a connected graph is a spanning tree with maximum degree at mostk. We show here that if\(t(G) \geqslant \frac{1}{{(k - 2)}}\), withk ≥ 3, thenG has ak-tree. The notion of ak-tree generalizes the casek = 2 of a hamiltonian path, so that this result, as we discuss, may be of some interest in connection with Chvátal's conjecture that, for somet, every graph with toughness at leastt is hamiltonian.
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References
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Win, Sein: Über Gerüste mit vorgeschriebenen Maximalgraden in Graphen und eine Vermutung von Las Vergnas. Ph.D. Thesis, Universität Hamburg (1979)
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Win, S. On a connection between the existence ofk-trees and the toughness of a graph. Graphs and Combinatorics 5, 201–205 (1989). https://doi.org/10.1007/BF01788671
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DOI: https://doi.org/10.1007/BF01788671