Hamiltonian decomposition of complete tripartite 3-uniform hypergraphs (Q2786889)

A complete tripartite 3-uniform hypergraph \(K_{m,m,m}^{(3)} \) has its vertex set \(V\) partitioned into three subsets \(V_1\), \(V_2\) and \(V_3\) of cardinality \(m\) and its edges are all 3-subsets of vertices which are not contained in any of the \(V_i\). A Hamiltonian cycle in a 3-uniform hypergraph is a cyclic ordering of its vertices such that every consecutive 3-tuple of vertices is an edge. The purpose of the paper under review is to show that the necessary condition for \(K_{m,m,m}^{(3)} \) to have a Hamiltonian decomposition, namely that \(m\) is divisible by 3, is also sufficient. The proof is constructive and uses 1-factors and orthogonal quasigroups.