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

TL;DR

This paper investigates specific types of 1-factorisations in complete 3-uniform hypergraphs, identifying conditions under which they are connected or uniform, and extends previous constructions with new existence results.

Contribution

It characterizes when the known family of 1-factorisations is connected or uniform, and identifies new cases where these properties hold, extending prior work.

Findings

Connected 1-factorisation only for q=2,5,11 or q=2^p with odd p

Uniform 1-factorisation only for q=2,5,8

Each identified case is also a uniform-connected 1-factorisation

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).