Detachments of Amalgamated 3-uniform Hypergraphs : Factorization Consequences

Abstract

A detachment of a hypergraph $\scr F$ is a hypergraph obtained from $\scr F$ by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph $\scr G$ can be thought of as taking $\scr G$, partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph $\scr F$. 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 $\scr G_1\cup \ldots \cup\scr G_k$ of $k$ edge-disjoint factors, where for $i=1,\dots ,k$, $\scr G_i$ 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) ${\sum }_{i=1}^k{r}_i=\lambda \left(\genfrac{}{}{0ex}{}{n-1}{2}\right)m^2$.

20 pages, 4 figures

Cited by in corpus (1)

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