Uniform coverings of 2-paths in the complete graph and the complete bipartite graph

Let G be a graph and H a subgraph of G. A D(G, H, λ) design is a collection V of subgraphs of G each isomorphic to H so that every 2-path (path of length 2) in G lies in exactly λsubgraphs in D. The problem of constructing D(Kn,Cn,l) designs is the so-called Dudeney's round table problem. We denote by Cκ a cycle on κ vertices and by p. a path on κ vertices. In this paper, we construct D(Kn,n, C2n, 1) designs and D(Kn, Pn,l) designs when n = 0,1,3 (mod'4); and D(Kn,n, C2n,2) designs and D(Kn, Pn, 2) designs when n = 2 (mod 4). The existence problems of D[Kn,n,C2n,l) designs and D{Kn,Pn,l) designs for n = 2 (mod 4) remain open.