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Let \(G\) be a graph and \(H\) a subgraph of \(G\). A \(D(G, H, \lambda)\) design is a collection \(\mathcal{D}\) of subgraphs of \(G\) each isomorphic to \(H\) so that every \(2\)-path (path of length \(2\)) in \(G\) lies in exactly \(\lambda\) subgraphs in \(\mathcal{D}\). The problem of constructing \(D(K_n,C_n,1)\) designs is the so-called Dudeney’s round table problem. We denote by \(C_k\), a cycle on \(k\) vertices and by \(P_k\), a path on \(k\) vertices.
In this paper, we construct \(D(K_{n,n},C_{2n},1)\) designs and \(D(K_n,P_n,1)\) designs when \(n \equiv 0,1,3 \pmod{4}\); and \(D(K_{n,n},C_{2n},2)\) designs and \(D(K_n,P_n,2)\) designs when \(n \equiv 2 \pmod{4}\). The existence problems of \(D(K_{n,n},C_{2n},1)\) designs and \(D(K_n,P_n,1)\) designs for \(n \equiv 2 \pmod{4}\) remain open.