A packing of \(K_n\) with copies of \(C_4\) (the cycle of length \(4\)), is an ordered triple \((V, \mathcal{C}, L)\), where \(V\) is the vertex set of the complete graph \(K_n\), \(C\) is a collection of edge-disjoint copies of \(C_4\), and \(L\) is the set of edges not belonging to a block of \(\mathcal{C}\). The number \(n\) is called the order of the packing and the set of unused edges \(L\) is called the leave. If \(C\) is as large as possible, then \((V, \mathcal{C}, L)\) is called a maximum packing MPC\((n, 4, 1)\). We say that an handcuffed design \(H(v, k, 1)\) \((W, P)\) is embedded into an MPC\((n, 4, 1)\) \((V, C, L)\) if \(W \subseteq V\) and there is an injective mapping \(f : \mathcal{P} \to \mathcal{C}\) such that \(P\) is a subgraph of \(f(P)\) for every \(P \in \mathcal{P}\). Let \(\mathcal{SH}(n, 4, k)\) denote the set of the integers \(v\) such that there exists an MPC\((n, 4, 1)\) which embeds an \(H(v, k, 1)\). If \(n \equiv 1 \pmod 8\) then an MPC\((n, 4, 1)\) coincides with a \(4\)-cycle system of order \(n\) and \(\mathcal{SH}(n, 4, k)\) is found by Milici and Quattrocchi, Discrete Math., \(174 (1997)\).
The aim of the present paper is to determine \(\mathcal{SH}(n, 4, k)\) for every integer \(n \not\equiv 1 \pmod 8\), \(n \geq 4\).
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