Connected Baranyai’s theorem

Abstract

Let K ^h _n = (V, ( ^V _h )) be the complete h-uniform hypergraph on vertex set V with ¦V¦ = n. Baranyai showed that K ^h _n can be expressed as the union of edge-disjoint r-regular factors if and only if h divides rn and r divides $(_{h - 1}^{n - 1} )$. Using a new proof technique, in this paper we prove that λK ^h _n can be expressed as the union $\mathcal{G}_1 \cup ... \cup \mathcal{G}_k$ of k edge-disjoint factors, where for 1≤i≤k, $\mathcal{G}_i$ is r _i -regular, if and only if (i) h divides r _i n for 1≤i≤k, and (ii) $\sum\nolimits_{i = 1}^k {r_i = \lambda (_{h - 1}^{n - 1} )}$. Moreover, for any i (1≤i≤k) for which r _i ≥2, this new technique allows us to guarantee that $\mathcal{G}_i$ is connected, generalizing Baranyai’s theorem, and answering a question by Katona.