-uniform friendship hypergraph is an r
-uniform hypergraph with the property for any set X of r
vertices there is a unique vertex cX
so that (X-v
} is a hyperedge for every vertex v
in X. cX
is called the completion of X.
A 2-uniform friendship hypergraph is exacly a friendship graph
Friendship hypergraphs were introduced (for r
=3) by Sós in
- V. T. Sós, Some remarks on the connection of graph theory, finite geometry and block designs.
Colloquio Internationale sulle Teorie Combinatorie, 223-233, (1976).
A vertex u
in a friendship hypergraph is a universal friend if every r
-set of vertices containing u
is a hyperedge.
Sós proved that an n
vertex 3-uniform friendship hypergraph
with a universal friend is equivalent to a Steiner triple system on n
This result is easily generalized to other values of r.
Some years later, the first five examples of 3-uniform friendship hypergraphs were found in:
- S. G. Hartke and J. Vandenbussche, On a question of Sós about 3-uniform friendship hypergraphs.
Journal Combinatorial Designs, vol. 16, 253-261 (2008).
We found an infinite family of 3-uniform friendship hypergraphs on 2k
=3,4,5, . . ., in:
- L. K. Jřrgensen and A. A. Sillasen, On the existence of friendship hypergraphs,
To appear in: Journal of Combinatorial Designs. DOI: 10.1002/jcd.21388
In this paper we also find 3-uniform friendship hypergraphs on 20 and 28 vertices.
And we find a 4-uniform friendship hypergraph on 9 vertices.
In this file
we list the edges of some friendship hypergraphs.