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Toughness and vertex degrees

Bauer, D., Broersma, H. J., van den Heuvel, J. ORCID: 0000-0003-0897-9148, 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
Date Deposited: 09 Apr 2010 13:16
Last Modified: 18 Jun 2024 20:21
Projects: EP/F064551/1
Funders: Engineering and Physical Sciences Research Council

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