an:06137926
Zbl 1258.05087
Bahmanian, M. Amin
Detachments of amalgamated 3-uniform hypergraphs: factorization consequences
EN
J. Comb. Des. 20, No. 11-12, 527-549 (2012).
00310870
2012
j
05C65 05C70
amalgamations; detachments; 3-uniform hypergraphs; laminar families; factorization; decomposition
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:
{\parindent=7mm
\begin{itemize}\item[(i)]\(m_{i}=m_j\) for all \(1\leq i,j \leq k\)
\item[(ii)]\(3 \mid r_{i}mn\) for each \(i\), \(1 \leq i \leq k\), and
\item[(iii)]\(\sum^{k}_{i=1} r_{i} =
\lambda \binom{n-1}{2} m^2 \).
\end{itemize}}