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It is known that the 3-uniform loose 3-cycle decomposes the complete 3-uniform hypergraph of order \(v\) if and only if \(v \equiv 0, 1,\text{ or }2 (\operatorname{mod} 9)\). For all positive integers \(\lambda\) and \(v\), we find a maximum packing with loose 3-cycles of the \(\lambda\)-fold complete 3-uniform hypergraph of order \(v\). We show that, if \(v \geq 6\), such a packing has a leave of two or fewer edges.more » « less
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A $$k$$-factorization of the complete $$t$$-uniform hypergraph $$K^{(t)}_{v}$$ is an $$H$$-decomposition of $$K^{(t)}_{v}$$ where $$H$$ is a $$k$$-regular spanning subhypergraph of $$K^{(t)}_{v}$$. We use nauty to generate the 2-regular and 3-regular spanning subhypergraphs of $$K^{(3)}_v$$ for $$v\leq 9$$ and investigate which of these subhypergraphs factorize $$K^{(3)}_v$$ or $$K^{(3)}_v-I$$, where $$I$$ is a 1-factor. We settle this question for all but two of these subhypergraphs.more » « less
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null (Ed.)Let $$G$$ be one of the two multigraphs obtained from $$K_4-e$$ by replacing two edges with a double-edge while maintaining a minimum degree of~2. We find necessary and sufficient conditions on $$n$$ and $$\lambda$$ for the existence of a $$G$$-decomposition of $$^{\lambda}K_n$$.more » « less
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null (Ed.)Consider the multigraph obtained by adding a double edge to $$K_4-e$$. Now, let $$D$$ be a directed graph obtained by orientating the edges of that multigraph. We establish necessary and sufficient conditions on $$n$$ for the existence of a $$(K^{*}_{n},D)$$-design for four such orientations.more » « less
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