Detachments of Amalgamated 3-uniform Hypergraphs : Factorization Consequences

TL;DR

This paper proves a new detachment theorem for 3-uniform hypergraphs using Nash-Williams lemma, enabling the decomposition of complete multipartite hypergraphs into edge-disjoint regular factors under specific conditions.

Contribution

It generalizes previous amalgamation theorems by establishing a detachment theorem for 3-uniform hypergraphs with broad factorization applications.

Findings

Established a detachment theorem for 3-uniform hypergraphs.

Applied the theorem to decompose complete multipartite hypergraphs.

Derived necessary and sufficient conditions for hypergraph factorizations.

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 $\lambda K_{m_1,\ldots,m_n}^{3}$ can be expressed as the union $\scr G_1\cup \ldots \cup\scr G_k$ of $k$ edge-disjoint factors, where for $i=1,\ldots, k$, $\scr G_i$ is $r_i$-regular, if and only if (i) $m_i=m_j:=m$ for all $1\leq i,j\leq k$, (ii) $3$ divides $r_imn$ for each $i$, $1\leq i\leq k$, and (iii) $\sum_{i=1}^{k} r_i=\lambda \binom{n-1}{2}m^2$.