Embeddings of One-Factorizations of Hypergraphs and Decompositions of Partitions

Abstract

We look at one-factorizations of complete k-uniform hypergraphs, and investigate the problem of determining when, for U a subset of V, one can embed a one-factorization of the complete k-uniform hypergraph on U in a one-factorization of the complete k-uniform hypergraph on V. We give a brief history of the problem, and find our own independent results for specific values of k and v = |V|, in the process making explicit a theorem implicitly used by Haggkvist and Hellgren in their solution to the problem in general. We provide our own independent proof of this theorem, and subsequently use it to extend our results to certain nonuniform hypergraphs. This, in particular, allows us to find alternate proofs about two results involving the extension of symmetric Latin squares, originally shown by Cruse and by Hoffman. We then explain the connection between the hypergraph-embedding problem and a problem involving the decomposition of partitions of an integer N into subpartitions of an integer n, where n divides N. This in turn leads to a problem involving the cone generated by a subset V of R^n, the properties of which we investigate thoroughly.