Abstract
The complete 3-uniform hypergraph of order v has a set V of size v as its vertex set and the set of all 3-element subsets of V as its edge set. The degree of a vertex is the number of edges in its edge set that contain it. We consider a class of 3-uniform hypergraphs with 5 edges and 10 vertices such that: every vertex has degree either 1 or 2 and any two edges intersect in at most one vertex. There are 5 such hypergraphs. For k in {1,2,3,4,5}, let Hk denote the hypergraphs with vertex set {v1, v2, v3, v4, v5, v6, v7, v8, v9, v10} and edge sets {{v1, v2, v3}, {v1, v4, v5}, {v2, v4, v6}, {v3, v7, v8}, {v5, v9, v10}}, {{v1, v2, v3}, {v1, v4, v5}, {v2, v4, v6},{v3, v7, v8}, {v7, v9, v10}}, {{v1, v2, v3}, {v1, v4, v5}, {v2, v6, v7}, {v3, v8, v9}, {v4, v6, v10}}, and {{v1, v2, v3}, {v1, v4, v5}, {v2, v6, v7}, {v4, v8, v9}, {v6, v8, v10}} respectively. We give necessary and sufficient conditions for the existence of a decomposition of the complete 3-uniform hypergraph of order v into isomorphic copies of each Hk.
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