Uniform coverings of 2-paths with 4-cycles

Let GG be a graph [a digraph] and HH be a subgraph of GG. A D(G,H,λ)D(G,H,λ)design   is a multiset DD of subgraphs of GG each isomorphic to HH so that every 2-path [directed 2-path] of GG lies in exactly λλ subgraphs in DD. In this paper, we show that there exists a D(Kn,n,C4,λ)D(Kn,n,C4,λ) design if and only if (i) nn is even, or (ii) nn is odd and λλ is even. We also show that there exists a D(Kn,n∗,C⃗4,λ) design for every nn and λλ, where Kn,nKn,n and Kn,n∗ are the complete bipartite graph and the complete bipartite digraph, respectively; C4C4 and C⃗4 are a 4-cycle and a directed 4-cycle, respectively.