# 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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## 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 $t\ge1$, 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 http://onlinelibrary.wiley.com/ © 2013 John Wiley & Sons, Inc. Q Science > QA Mathematics Departments > Mathematics degree sequences; toughness; best monotone theorem Engineering and Physical Sciences Research Council EP/F064551/1 09 Apr 2010 13:16 http://eprints.lse.ac.uk/27680/