Cayley, Marty and Schreier hypergraphs (Q1341254)

In this paper, Cayley, Marty and Schreier hypergraphs are defined. Many properties of these hypergraphs are proved. Typical results are: (i) Every projective plane \(\text{PG} (2,q)\) with \(q^ 2 + q + 1\) a prime, is a connected and sharply vertex-transitive hypergraph of rank \(q + 1\) but is not a \((q + 1)\)-Cayley hypergraph (G. Sabidussi proved in 1958 that any connected and sharply vertex-transitive graph is a Cayley graph); (ii) Every simple hypergraph \(H\) is a generalized Schreier hypergraph. In the last two sections, a characterization of linear spaces in terms of generalized Schreier hypergraphs and a relation between a connected graph admitting a regular \(k\)-edge-colouring and a Cayley graph are proved.