Generalizations of Cayley graphs to uniform hypergraphs

A vertex-transitive graph is a Cayley graph if and only if it admits a group of automorphisms acting regularly on its set of vertices. We generalize Cayley graphs to k-uniform hypergraphs, i.e. pairs (V, ℰ) where ℰ consists of k-subsets of V, k ≥ 2. We call a k-uniform hypergraph Cayley if and only if the hypergraph admits a group of automorphisms acting regularly on the vertex set of the hypergraph. We compare our definition with two other generalizations of Cayley graphs to hypergraphs found in the literature, and argue that ours is the most general; the other generalizations are special cases of the one proposed in this article. We present some basic properties of Cayley hypergraphs, discuss the connection of this concept to some recent results concerning hypergraphs, as well as connections to our ongoing project of constructing k-uniform hypergraphs whose full automorphism group is isomorphic to a prescribed finite group G and acts regularly on the vertices of the hypergraph.