Embedding connected factorizations

Abstract

For r:=(r1,…,rq), an r-factorization of the complete λ-fold h-uniform m-vertex hypergraph λKmh is a partition of the edges of λKmh into F1,…,Fq such that each color class Fi is ri-regular and spanning. We prove two results on embedding factorizations. Previously, these results were only known for a few small values of h, and even then only partially. We show that for n⩾hm and s:=(s1,…,sk), the obvious necessary conditions that ensure that an r-factorization of λKmh can be embedded into an s-factorization of λKnh are also sufficient. This extends Cruse's theorem, Baranyai's theorem, and Häggkvist-Hellgren's theorem. A connectedr-factorization is an r-factorization in which each color class is connected. For n⩾hm, we establish the necessary and sufficient conditions under which an r-factorization of λKmh can be embedded into a connected s-factorization of λKnh. This extends Walecki's theorem, and Hilton's theorem on embedding Hamiltonian decompositions (take λ=r1=…=rq=1,h=s1=…=sk=2).