Decompositions of complete uniform hypergraphs into Hamilton Berge cycles

In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if $n$ divides $\binom{n}{k}$, 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 $v_1,e_1,v_2,\dots,v_n,e_n$ 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 $k \ge 4$ and $n \ge 30$. Our argument is based on the Kruskal-Katona theorem. The case when $k=3$ was already solved by Verrall, building on results of Bermond

Decompositions of Complete Uniform Multipartite Hypergraphs

In recent years, researchers have studied the existence of complete uniform hypergraphs into small-order hypergraphs. In particular, results on small 3-uniform graphs including loose 3, 4, and 5 cycles have been studied, as well as 4-uniform loose cycles of length 3. As part of these studies, decompositions of multipartite hypergraphs were constructed. In this paper, we extend this work to higher uniformity and order as well as expand the class of hypergraphs