On the minimum degree of minimally $t$-tough, claw-free graphs

Abstract

A graph $G$ is minimally $t$-tough if the toughness of $G$ is $t$ and deletion of any edge from $G$ decreases its toughness. Katona et al. conjectured that the minimum degree of any minimally $t$-tough graph is $\lceil 2t\rceil$ and proved that the minimum degree of minimally $\frac{1}{2}$-tough and $1$-tough, claw-free graphs is 1 and 2, respectively. We have show that every minimally $3/2$-tough, claw-free graph has a vertex of degree of $3$. In this paper, we give an upper bound on the minimum degree of minimally $t$-tough, claw-free graphs for $t\ge 2$.