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Detachments of amalgamated 3-uniform hypergraphs: factorization consequences. (English) Zbl 1258.05087
Summary: A detachment of a hypergraph $$\mathcal F$$ is a hypergraph obtained from $$\mathcal F$$ by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph $$\mathcal G$$ can be thought of as taking $$\mathcal G$$, partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph $$\mathcal F$$.
In this paper, we use Nash-Williams lemma on laminar families to prove a detachment theorem for amalgamated 3-uniform hypergraphs, which yields a substantial generalization of previous amalgamation theorems by Hilton, Rodger, and Nash-Williams.
To demonstrate the power of our detachment theorem, we show that the complete 3-uniform $$n$$-partite multihypergraph $$\lambda K^{3}_{m_{1},\dots ,m_n}$$ can be expressed as the union $$\mathcal G_{1} \cup \dots \cup \mathcal G_{k}$$ of $$k$$ edge-disjoint factors, where for $$i=1,\dots,k$$, $$\mathcal G_{i}$$ is $$r_i$$-regular, if and only if:
(i)
$$m_{i}=m_j$$ for all $$1\leq i,j \leq k$$
(ii)
$$3 \mid r_{i}mn$$ for each $$i$$, $$1 \leq i \leq k$$, and
(iii)
$$\sum^{k}_{i=1} r_{i} = \lambda \binom{n-1}{2} m^2$$.

##### MSC:
 05C65 Hypergraphs 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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