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arXiv · 2307.13196

An Infinite Family of Connected 1-Factorisations of Complete 3-Uniform Hypergraphs

Barbara Maenhaut·Jeremy Mitchell·Anna Puskas
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Abstract

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 $q\equiv 2\pmod 3$ 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).

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Barbara Maenhaut, Jeremy Mitchell, Anna Puskas. 2023-07-25. An Infinite Family of Connected 1-Factorisations of Complete 3-Uniform Hypergraphs. https://arxiv.org/abs/2307.13196

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