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 $.
Categories: math.CO