Decompositions of complete uniform hypergraphs into Hamilton Berge cycles

摘要

In 1973 Bermond, Germa, Heydemann and Sotteau conjec- tured that if n divides (_k~n), then the complete k-uniform hy- pergraph on n vertices has a decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an alternating se- quence 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 divis- ibility 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.