Embedding factorizations for 3-uniform hypergraphs

TL;DR

This paper introduces new conditions for embedding 3-uniform hypergraphs with colored hyperedges into larger hypergraph factorizations, using novel amalgamation techniques to address longstanding questions in hypergraph theory.

Contribution

It provides the first use of amalgamation techniques for hypergraph embedding and establishes necessary and sufficient conditions for embedding colored hypergraphs with prescribed pieces.

Findings

Derived conditions for embedding hyperedge-colored complete 3-uniform hypergraphs.

Extended embedding conditions to hyperedges with prescribed pieces.

Progress towards solving Cameron's problem on hypergraph 1-factorizations.

Abstract

In this paper, two results are obtained on a hypergraph embedding problem. The proof technique is itself of interest, being the first time amalgamations have been used to address the embedding of hypergraphs. The first result finds necessary and sufficient conditions for the embedding a hyperedge-colored copy of the complete 3-uniform hypergraph of order $m$, $K_m^3$, into an $r$-factorization of $K_n^3$, providing that $n> 2m+(-1+\sqrt{8m^2-16m-7})/2$. The second result finds necessary and sufficient conditions for an embedding when not only are the colors of the hyperedges of $K_m^3$ given, but also the colors of all the "pieces" of hyperedges on these $m$ vertices are prescribed (the "pieces" of hyperedges are eventually extended to hyperedges of size 3 in $K_n^3$ by adding new vertices to the hyperedges of size 1 and 2 during the embedding process). Both these results make progress towards settling an old question of Cameron on completing partial 1-factorizations of hypergraphs.