An Infinite Family of Connected 1-Factorisations of Complete 3-Uniform Hypergraphs - Maenhaut, Barbara; Mitchell, Jeremy; Puskás, Anna (The Electronic Journal of Combinatorics, 2024)
A connected 1-factorisation is a 1-factorisation of a hypergraph for which the union of each pair of distinct 1-factors is a connected hypergraph. A uniform 1-factorisation is a 1-factorisation of a hypergraph for which the union of each pair of distinct 1-factors is isomorphic to the same subhypergraph, and a uniform-connected 1-factorisation is a uniform 1-factorisation in which that subhypergraph is connected. Chen and Lu [Journal of Algebraic Combinatorics, 46(2) 475–497, 2017] describe a family of 1-factorisations of the complete 3-uniform hypergraph on q+1 vertices, where q23 is a prime power. In this paper, we show that their construction yields a connected 1-factorisation only when q=2,5,11 or q=2^p for some odd prime p, and a uniform 1-factorisation only for q=2,5,8 (each of these is a uniform-connected 1-factorisation).