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Embedding factorizations for 3-uniform hypergraphs II: $$r$$-factorizations into $$s$$-factorizations. (English) Zbl 1337.05084
Summary: Motivated by a 40-year-old problem due to P. J. Cameron [Parallelisms of complete designs. Cambridge etc.: Cambridge University Press (1976; Zbl 0333.05007)] on extending partial parallelisms, we provide necessary and sufficient conditions under which one can extend an $$r$$-factorization of a complete $$3$$-uniform hypergraph on $$m$$ vertices, $$K_m^3$$, to an $$s$$-factorization of $$K_n^3$$. This generalizes an existing result of ZS. Baranyai and A. E. Brouwer [Extension of colourings of the edges of a complete (uniform hyper) graph. Amsterdam: Math. Centrum (1977; Zbl 0362.05059)] – where they proved it for the case $$r=s=1$$.
For Part I see [A. Bahmanian and C. Rodger, J. Graph Theory 73, No. 1–2, 216–224 (2013; Zbl 1264.05088)].

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C65 Hypergraphs 05C15 Coloring of graphs and hypergraphs 05B40 Combinatorial aspects of packing and covering 05B05 Combinatorial aspects of block designs
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