  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\geq 2 $.
