jmc

The $6$ green marbles and yellow marbles form $1-\frac{1}{3}-\frac{1}{4}=\frac{5}{12}$ of the total marbles. Now, suppose the total number of marbles is $x$. We know the number of yellow marbles is $\frac{5}{12}x-6$ and a positive integer. Therefore, $12$ must divide $x$. Trying the smallest multiples of $12$ for $x$, we see that when $x=12$, we get there are $-1$ yellow marbles, which is impossible. However when $x=24$, there are $\frac{5}{12}\cdot 24-6=\boxed{4}$ yellow marbles, which must be the smallest possible.