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≤ik, \(\mathcal{G}_i \) is r i -regular, if and only if (i) h divides r i n for 1≤ik, and (ii) \(\sum\nolimits_{i = 1}^k {r_i = \lambda (_{h - 1}^{n - 1} )} \). Moreover, for any i (1≤ik) 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.

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Correspondence to M. Amin Bahmanian.

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Bahmanian, M.A. Connected Baranyai’s theorem. Combinatorica 34, 129–138 (2014). https://doi.org/10.1007/s00493-014-2928-3

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Mathematics Subject Classification (2000)

  • 05C15
  • 05C40
  • 05C51
  • 05C65
  • 05C70
  • 05B40
  • 05B05