Detachments of Amalgamated 3-uniform Hypergraphs : Factorization Consequences

A detachment of a hypergraph $\text{scr}F$ is a hypergraph obtained from $\text{scr}F$ by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph $\text{scr}G$ can be thought of as taking $\text{scr}G$, partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph $\text{scr}F$. 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 multi-hypergraph $\lambda K_{m_1,\dots ,m_n}^3$ can be expressed as the union $\text{scr}{G}_1\cup \dots \cup \text{scr}{G}_k$ of $k$ edge-disjoint factors, where for $i=1,\dots ,k$, $\text{scr}{G}_i$ is $r_i$-regular, if and only if (i) $m_i=m_j:=m$ for all $1\le i,j\le k$, (ii) $3$ divides $r_{i}mn$ for each $i$, $1\le i\le k$, and (iii) ${\sum }_{i=1}^k{r}_i=\lambda \left(\genfrac{}{}{0px}{}{n-1}{2}\right)m^2$.