Geodesic
An infinite family of connected 1-factorisations of complete 3-uniform hypergraphs
Barbara Maenhaut1 ; Jeremy Mitchell1 ; Anna Puskás2
1The University of Queensland
2University of Glasgow
The electronic journal of combinatorics, Tome 31 (2024) no. 4
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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).
DOI : 10.37236/12283
Classification : 05C70, 05C65, 05E18, 05C75
Mots-clés : symmetric factorization, \(k\)-homogeneous permutation group, fractional linear mapping

Barbara Maenhaut  1   ; Jeremy Mitchell  1   ; Anna Puskás  2

1 The University of Queensland
2 University of Glasgow 
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Barbara  Maenhaut; Jeremy Mitchell;  Anna Puskás. An infinite family of connected 1-factorisations of complete 3-uniform hypergraphs. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12283

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