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Toughness and vertex degrees

Bauer, D., Broersma, H. J., van den Heuvel, J., Kahl, N. and Schmeichel, E. (2013) Toughness and vertex degrees. Journal of Graph Theory, 72 (2). pp. 209-219. ISSN 0364-9024

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Identification Number: 10.1002/jgt.21639


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 $t\ge1$, but then show that for any integer $k\ge1$, a best monotone theorem for $t=\frac1k\le 1$ requires at least $f(k)\cdot|V(G)|$ nonredundant conditions, where $f(k)$ grows superpolynomially as $k\rightarrow\infty$. When $t<1$, we give an additional, simple theorem for $G$ to be $t$-tough, in terms of its vertex degrees.

Item Type: Article
Official URL:
Additional Information: © 2013 John Wiley & Sons, Inc.
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Sets: Departments > Mathematics
Date Deposited: 09 Apr 2010 13:16
Last Modified: 20 Jan 2020 04:48
Projects: EP/F064551/1
Funders: Engineering and Physical Sciences Research Council

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