# Toughness and vertex degrees

D. Bauer, Haitze J. Broersma, J. van den Heuvel, N. Kahl, E. Schmeichel

3 Citations (Scopus)

### Abstract

We study theorems giving sufficient conditions on the vertex degrees of a graph G to guarantee G is t-tough. We first give a best monotone theorem when t is at least 1, but then show that for any integer k > 0, a best monotone theorem for t=1/k requires at least f(k)|V(G)| nonredundant conditions, where f(k) grows superpolynomially as k grows to infinity. When t < 1, we give an additional, simple theorem for G to be t-tough, in terms of its vertex degrees.
Original language Undefined 209-219 11 Journal of graph theory 72 2 https://doi.org/10.1002/jgt.21639 Published - 2013

### Keywords

• MSC-05C
• Toughness
• EWI-23370
• IR-86131
• Degree condition
• METIS-297654
• Best monotone theorem

### Cite this

Bauer, D., Broersma, H. J., van den Heuvel, J., Kahl, N., & Schmeichel, E. (2013). Toughness and vertex degrees. Journal of graph theory, 72(2), 209-219. https://doi.org/10.1002/jgt.21639
Bauer, D. ; Broersma, Haitze J. ; van den Heuvel, J. ; Kahl, N. ; Schmeichel, E. / Toughness and vertex degrees. In: Journal of graph theory. 2013 ; Vol. 72, No. 2. pp. 209-219.
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Bauer, D, Broersma, HJ, van den Heuvel, J, Kahl, N & Schmeichel, E 2013, 'Toughness and vertex degrees', Journal of graph theory, vol. 72, no. 2, pp. 209-219. https://doi.org/10.1002/jgt.21639

Toughness and vertex degrees. / Bauer, D.; Broersma, Haitze J.; van den Heuvel, J.; Kahl, N.; Schmeichel, E.

In: Journal of graph theory, Vol. 72, No. 2, 2013, p. 209-219.

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AU - Bauer, D.

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AU - van den Heuvel, J.

AU - Kahl, N.

AU - Schmeichel, E.

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KW - Toughness

KW - EWI-23370

KW - IR-86131

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Bauer D, Broersma HJ, van den Heuvel J, Kahl N, Schmeichel E. Toughness and vertex degrees. Journal of graph theory. 2013;72(2):209-219. https://doi.org/10.1002/jgt.21639