# Toughness and vertex degrees

Bauer, D., Broersma, H. J., van den Heuvel, J., Kahl, N. and Schmeichel, E. (2009) Toughness and vertex degrees. arXiv.

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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: Monograph (Other) http://arxiv.org/ © 2009 The authors Q Science > QA Mathematics Departments > Mathematics http://www.lse.ac.uk/library/rights/LSERO.htm http://eprints.lse.ac.uk/27680/