Uniform coverings of 2-paths with 4-cycles

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