H-Decomposition of r-Graphs when H is an r-Graph with Exactly k Independent Edges
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Let \(\phi _H^r(n)\) be the smallest integer such that, for all r-graphs G on n vertices, the edge set E(G) can be partitioned into at most \(\phi _H^r(n)\) parts, of which every part either is a single edge or forms an r-graph isomorphic to H. The function \(\phi ^2_H(n)\) has been well studied in literature, but for the case \(r\ge 3\), the problem of determining \(\phi _H^r(n)\) is widely open. Sousa (Electron J Comb 17:R40, 2010) gave an asymptotic value of \(\phi _H^r(n)\) when H is an r-graph with exactly 2 edges, and determined the exact value of \(\phi _H^r(n)\) in some special cases. In this paper, we give the exact value of \(\phi _H^r(n)\) when H is an r-graph with exactly 2 edges, which completes Sousa’s result, we further determine the exact value of \(\phi _H^r(n)\) when H is an r-graph consisting of exactly k independent edges.
KeywordsHypergraph Decomposition Independent edges
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