Embedding Irregular Colorings into Connected Factorizations

For $r:=(r1,\dots,rk)$, an $r$-factorization of the complete $\lambda$-fold $h$-uniform $n$-vertex hypergraph $\lambda Kn^h$ is a partition of (the edges of) $\lambda Kn^h$ into $F1,\dots, Fk$ such that for $i=1,\dots,k$, $Fi$ is $ri$-regular and spanning. Suppose that $n \geq (h-1)(2m-1)$. Given a partial $r$-factorization of $\lambda Km^h$, that is, a coloring (i.e. partition) $P$ of the edges of $\lambda Km^h$ into $F1,\dots, Fk$ such that for $i=1,\dots,k$, $Fi$ is spanning and the degree of each vertex in $Fi$ is at most $ri$, we find necessary and sufficient conditions that ensure $P$ can be extended to a connected $r$-factorization of $\lambda Kn^h$ (i.e. an $r$-factorization in which each factor is connected). Moreover, we prove a general result that implies the following. Given a partial $s$-factorization $P$ of any sub-hypergraph of $\lambda Km^h$, where $s:=(s1,\dots,sq)$ and $q$ is not too big, we find necessary and sufficient conditions under which $P$ can be embedded into a connected $r$-factorization of $\lambda Kn^h$. These results can be seen as unified generalizations of various classical combinatorial results such as Cruse's theorem on embedding partial symmetric latin squares, Baranyai's theorem on factorization of hypergraphs, Hilton's theorem on extending path decompositions into Hamiltonian decompositions, H\"{a}ggkvist and Hellgren's theorem on extending 1-factorizations, and Hilton, Johnson, Rodger, and Wantland's theorem on embedding connected factorizations.