Properties of minimally t-tough graphs

Gyula Y. Katona, Dániel Soltész, Kitti Varga
2018 Discrete Mathematics  
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) ≤ n 3 + 1. We also prove that every minimally 1-tough, claw-free graph is a cycle. On the other hand, we show that for every positive rational number t any graph can be embedded as an induced subgraph into a minimally t-tough graph.
doi:10.1016/j.disc.2017.08.033 fatcat:qjafkqrx35hbnn2sk5z7kpc7t4