Embedding connected factorizations II

Abstract

Let $\lambda K_n^h$ be the complete h-uniform n-vertex hypergraph in which each edge is repeated λ times. For $r:=(r_1,\dots ,r_k)$, a (partial) r-factorization of $\lambda K_n^h$ is a partition of the edges of $\lambda K_n^h$ into factors $F_1,\dots ,F_k$ such that each factor is spanning and the degree of all vertices in each $F_i$ is (at most) $r_i$. Suppose that $n\ge (h-1)(2m-1)$. We establish necessary and sufficient conditions that ensure a partial r-factorization of $\lambda K_m^h$ can be embedded in a connected r-factorization of $\lambda K_n^h$. Moreover, we prove a general result which leads to a complete characterization of partial $(s_1,\dots ,s_q)$-factorizations of any sub-hypergraph of $\lambda K_m^h$ in connected r-factorizations of $\lambda K_n^h$ so long as q meets a natural upper bound. These results can be seen as unified generalizations of many classical combinatorial results, and can also be restated as results on embedding partial symmetric latin hypercubes.