Decompositions of complete uniform hypergraphs into Hamilton Berge cycles

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if n divides ( n k ) , then the complete k-uniform hypergraph on n vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an alternating sequence v 1 , e 1 , v 2 , … , v n , e n of distinct vertices v i and distinct edges e i so that each e i contains v i and v i + 1 . So the divisibility condition is clearly necessary. In this note, we prove that the conjecture holds whenever k ≥ 4 and n ≥ 30 . Our argument is based on the Kruskal–Katona theorem. The case when k = 3 was already solved by Verrall, building on results of Bermond.