*H*-Decomposition of *r*-Graphs when *H* is an *r*-Graph with Exactly *k* Independent Edges

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## Abstract

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.

## Keywords

Hypergraph Decomposition Independent edges## References

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