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 D of subgraphs of G each isomorphic to H so that every 2-path (path of length 2) in G lies in exactly λ subgraphs in D. The problem of constructing D(Kn, Cn, 1) designs is the so-called Dudeney’s round table problem. We denote by Ck a cycle on k vertices and by Pk a path on k vertices. In this paper, we construct D(Kn,n, C2n, 1) designs and D(Kn, Pn, 1) designs when n ≡ 0, 1, 3 (mod 4); andD(Kn,n, C2n, 2) designs andD(Kn, Pn, 2) designs when n ≡ 2 (mod 4). The existence problems of D(Kn,n, C2n, 1) designs and D(Kn, Pn, 1) designs for n ≡ 2 (mod 4) remain open.