Embedding arbitrary edge-colorings of hypergraphs into regular colorings

TL;DR

This paper establishes conditions under which partial edge-colorings of hypergraphs can be extended to complete factorizations, generalizing the concept of regular colorings in hypergraph theory.

Contribution

It provides a necessary and sufficient condition for extending partial hypergraph factorizations to larger complete hypergraphs, advancing hypergraph factorization theory.

Findings

Extension of partial factorizations is possible under specific size conditions.

Necessary conditions for hypergraph factorization extension are characterized.

Results apply to arbitrary edge-colorings of hypergraphs.

Abstract

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