Properties of minimally $t$-tough graphs
Gyula Y. Katona, Dániel Soltész, Kitti Varga · Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) · 2016
A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and the deletion of any edge from $G$ decreases the toughness. Kriesell conjectured that for every minimally $1$-tough graph the minimum degree $δ(G)=2$. We show that in every minimally $1$-tough graph $δ(G)\le\frac{n+2}{3}$. We also prove that every minimally $1$-tough claw-free graph is a cycle. On the other hand, we show that for every $t \in \mathbb{Q}$ any graph can be embedded as an induced subgraph into a minimally $t$-tough graph.