A \((\lambda K_n, G)\)-design is a partition of the edges of \(\lambda K_n\), into sub-graphs each of which is isomorphic to \(G\). In this paper, we investigate the existence of \((K_n, G_{16})\)-design and \((K_n, G_{20})\)-design, and prove that the necessary conditions for the existence of the two classes of graph designs are also sufficient.