by C. P. (Ben) Li, N. Singhi and G.H.J. van Rees
Abstract: The well-known Friendship Graph Theorem states that if a graph in which every pair of vertices has exactly one common neighbour, then G has a single vertex joined to all others, “a universal friend”. This has a beautiful proof that we present. So the problem of finding all friendship graphs is solved. So many researchers generalized the definition of a friendship graph. We will review some of these. Of particular interest to us is the generalization due to V. Sos. She defined the following friendship property for 3-uniform hypergraphs (every edge has 3 vertices). For every three vertices, x, y and z there exists a unique vertex w such that xyw, yzw and xzw are all edges in the 3-hypergraph. Her constructions featured “a universal friend”. Hartke and Vandenbussche showed constructions for 3-uniform hypergraphs on 8, 16 and 32 vertices. We improve the bounds on the size of a 3-uniform friendship hypergraphs. We show there are connections to geometry. We prove that the three 3-uniform hypergraphs on 16 points are geometrical and are the only geometrical 3-uniform hypergraphs on 16 points.
Biography: Professor G. H. J. van Rees took his 3 degrees at the University of Waterloo with his Ph.D. under R.C. Mullin being finished in 1978. He spent one year at the Math Department of Dalhousie University before joining Ralph Stanton at the Dept. of Computer Science where he has been ever since. He has written approximately 70 papers in refereed venues and is a full professor in his department. His research field is Combinatorics, specializing in Combinatorial Designs, Combinatorial Computing, Coding Theory, Latin Squares, and Lotto Designs.