Connected Baranyai's Theorem

TL;DR

This paper generalizes Baranyai's theorem by providing a new proof technique that characterizes how multiple edge-disjoint regular factors can decompose scaled complete hypergraphs, ensuring connectivity under certain conditions.

Contribution

The paper introduces a new proof method for decomposing scaled complete hypergraphs into regular factors, extending Baranyai's theorem and ensuring connectivity for factors with degree at least two.

Findings

Generalizes Baranyai's theorem to multiple factors

Provides conditions for decomposition of scaled hypergraphs

Guarantees connectivity of factors with degree ≥ 2

Abstract

Let $K_n^h=(V,\binom{V}{h})$ be the complete $h$-uniform hypergraph on vertex set $V$ with $|V|=n$. Baranyai showed that $K_n^h$ can be expressed as the union of edge-disjoint $r$-regular factors if and only if $h$ divides $rn$ and $r$ divides $\binom{n-1}{h-1}$. Using a new proof technique, in this paper we prove that $\lambda K_n^h$ can be expressed as the union $\mathcal G_1\cup \ldots \cup\mathcal G_k$ of $k$ edge-disjoint factors, where for $1\leq i\leq k$, $\mathcal G_i$ is $r_i$-regular, if and only if (i) $h$ divides $r_in$ for $1\leq i\leq k$, and (ii) $\sum_{i=1}^k r_i=\lambda \binom{n-1}{h-1}$. Moreover, for any $i$ ($1\leq i\leq k$) for which $r_i\geq 2$, this new technique allows us to guarantee that $\mathcal G_i$ is connected, generalizing Baranyai's theorem, and answering a question by Katona.