Perfect $1$-factorisations of complete $k$-uniform hypergraphs

A 1-factorisation of a graph is called perfect if it satisfies each of the following equivalent conditions: the union of each pair of 1-factors is isomorphic to the same connected subgraph, the union of each pair of 1-factors is connected, and the union of each pair of 1-factors is a Hamilton cycle. A 1-factorisation of a graph is called uniform if the union of each pair of 1-factors is isomorphic to the same subgraph. In this paper, we generalise the concept of uniform 1-factorisations from graphs to hypergraphs in the natural way, and, based on the three conditions above, we define four generalisations of perfect 1-factorisations of graphs to the context of hypergraphs (called connected-uniform, connected, Hamilton (cid:2) , and Hamilton Berge 1-factorisations). We then ask, for which values of k and n does the complete k -uniform hypergraph K kn admit such 1-factorisations. We show that, for k ≥ 3, uniform and uniform-connected 1-factorisations of complete k -uniform hypergraphs can exist only when k = 3, and when they exist they can be used to construct biplanes. We also show that, for k ≥ 2, all 1-factorisations of K k 2 k and K k 3 k are connected 1-factorisations, and prove the existence of non-connected 1-factorisations of K kmk for every m ≥ 4. We prove that Hamilton (cid:2) 1-factorisations of complete k -uniform hypergraphs do not exist for k ≥ 3. We then prove that, for k ≥ 2, all 1-factorisations of K k 2 k are Hamilton Berge 1-factorisations, and demonstrate