  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 gave some upper bounds on the minimum degree of the
minimally $ t $-tough graphs in \cite{Katona, Gyula}. In this paper, we show
that a minimally 1-tough graph $ G $ with girth $ g\geq 5 $ has minimum degree
at most $ \lfloor\frac{n}{g+1}\rfloor+g-1$, and a minimally $ 1 $-tough graph
with girth $ 4 $ has minimum degree at most $ \frac{n+6}{4}$. We also prove
that the minimum degree of minimally $\frac{3}2$-tough claw-free graphs is $ 3
$.
