Connected Baranyai's Theorem

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.