Embedding arbitrary edge-colorings of hypergraphs into regular colorings

Abstract

For $\text{r}=(r_1,\dots ,r_k)$, an $\text{r}$-factorization of the complete $\hat{I}»$-fold $h$-uniform $n$-vertex hypergraph $\hat{I}»K_n^h$ is a partition of the edges of $\hat{I}»K_n^h$ into $F_1,\dots ,F_k$ such that $F_j$ is $r_j$-regular and spanning for $1\le j\le k$. This paper shows that for $n>\frac{m-1}{1-2^{\frac{1}{1-h}}}+h-1$, a partial $\text{r}$-factorization of $\hat{I}»K_m^h$ can be extended to an $\text{r}$-factorization of $\hat{I}»K_n^h$ if and only if the obvious necessary conditions are satisfied.