Embedding factorizations for 3-uniform hypergraphs II: \(r\)-factorizations into \(s\)-factorizations. (English) Zbl 1337.05084

Bahmanian, Amin; Newman, Mike

Electron. J. Comb. 23, No. 2, Research Paper P2.42, 14 p. (2016).

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

Keywords:

factorizations; embedding; detachments; amalgamations; edge colorings; hypergraphs