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

## 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 |
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Official URL: | http://onlinelibrary.wiley.com/ |

Additional Information: | © 2013 John Wiley & Sons, Inc. |

Library of Congress subject classification: | Q Science > QA Mathematics |

Sets: | Departments > Mathematics |

Identification Number: | degree sequences; toughness; best monotone theorem |

Funders: | Engineering and Physical Sciences Research Council |

Projects: | EP/F064551/1 |

Date Deposited: | 09 Apr 2010 13:16 |

URL: | http://eprints.lse.ac.uk/27680/ |

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