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Detachments of Amalgamated 3-Uniform Hypergraphs Factorization Consequences



Amath formula is a hypergraph obtained from math formula by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph math formula can be thought of as taking math formula, partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph math formula. 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 math formula can be expressed as the union math formula of k edge-disjoint factors, where for math formula, math formula is math formula-regular, if and only if:

  1. math formula for all math formula,
  2. math formula for each i, math formula, and
  3. math formula.