Let ϕHr(n)\documentclass[12pt]{minimal}
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\begin{document}$$\phi _H^r(n)$$\end{document} 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 ϕHr(n)\documentclass[12pt]{minimal}
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\begin{document}$$\phi _H^r(n)$$\end{document} parts, of which every part either is a single edge or forms an r-graph isomorphic to H. The function ϕH2(n)\documentclass[12pt]{minimal}
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\begin{document}$$\phi ^2_H(n)$$\end{document} has been well studied in literature, but for the case r≥3\documentclass[12pt]{minimal}
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\begin{document}$$r\ge 3$$\end{document}, the problem of determining ϕHr(n)\documentclass[12pt]{minimal}
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\begin{document}$$\phi _H^r(n)$$\end{document} is widely open. Sousa (Electron J Comb 17:R40, 2010) gave an asymptotic value of ϕHr(n)\documentclass[12pt]{minimal}
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\begin{document}$$\phi _H^r(n)$$\end{document} when H is an r-graph with exactly 2 edges, and determined the exact value of ϕHr(n)\documentclass[12pt]{minimal}
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\begin{document}$$\phi _H^r(n)$$\end{document} in some special cases. In this paper, we give the exact value of ϕHr(n)\documentclass[12pt]{minimal}
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\begin{document}$$\phi _H^r(n)$$\end{document} when H is an r-graph with exactly 2 edges, which completes Sousa’s result, we further determine the exact value of ϕHr(n)\documentclass[12pt]{minimal}
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\begin{document}$$\phi _H^r(n)$$\end{document} when H is an r-graph consisting of exactly k independent edges.
