Toughness and Vertex Degrees

D. Bauer, H. J. Broersma, J. van den Heuvel, N. Kahl, E. Schmeichel

Published 2009-12-15, updated 2011-05-25Version 2

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.