Let 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(K n,n , C 4 , 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(K * n,n , − → C 4 , λ) design for every n and λ, where K n,n and K * n,n are the complete bipartite graph and the complete

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... artite digraph, respectively; C 4 and − → C 4 are a 4-cycle and a directed 4-cycle, respectively. c ⃝ 2015 Kalasalingam University. Production and Hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).