Embedding Irregular Colorings into Connected Factorizations

TL;DR

This paper establishes necessary and sufficient conditions for extending partial hypergraph factorizations into connected full factorizations, unifying several classical combinatorial theorems.

Contribution

It provides a general framework for extending partial hypergraph factorizations into connected factorizations, generalizing multiple classical results.

Findings

Conditions for extending partial factorizations into connected ones.

Unified approach encompassing classical theorems.

Generalization of embedding and extension results in hypergraph theory.

Abstract

For $r:=(r_1,\dots,r_k)$, an $r$-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 $F_1,\dots, F_k$ such that for $i=1,\dots,k$, $F_i$ is $r_i$-regular and spanning. Suppose that $n \geq (h-1)(2m-1)$. Given a partial $r$-factorization of $\lambda K_m^h$, that is, a coloring (i.e. partition) $P$ of the edges of $\lambda K_m^h$ into $F_1,\dots, F_k$ such that for $i=1,\dots,k$, $F_i$ is spanning and the degree of each vertex in $F_i$ is at most $r_i$, we find necessary and sufficient conditions that ensure $P$ can be extended to a connected $r$-factorization of $\lambda K_n^h$ (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 $\lambda K_m^h$, where $s:=(s_1,\dots,s_q)$ and $q$ is not too big, we find necessary and sufficient conditions under which $P$ can be embedded into a connected $r$-factorization of $\lambda K_n^h$. 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.