Consider two arbitrary polynomials $ p(x) $ and $ q(x) $ with highest degree terms $ x^n $ and $ x^m $, respectively. Then $ p(q(x)) = (q(x))^n + \cdots = (x^m + \cdots)^n + \cdots = x^{mn} + \cdots $ is a polynomial of degree $ mn $. It follows that $ f(g(x)) $ is a polynomial of degree $ 18 $. Then, either $ g(h(x)) $ or $ h(f(x)) $ must be a polynomial of degree $ 36 $. This gives that the degree of $ h(x) $ is either $ 12 $ or $ 6 $, but in the former case, the degree of $ h(f(x)) $ would be $ 72 $. Thus, the degree of $ h $ is $ \boxed{6} $.