Bounding the number of hyperedges in friendship r-hypergraphs

For r ≥ 2, an r-uniform hypergraph is called a friendship r-hypergraph if every set R of r vertices has a unique ‘friend’ – that is, there exists a unique vertex x /∈ R with the property that for each subset A ⊆ R of size r − 1, the set A ∪ {x} is a hyperedge. We show that for r ≥ 3, the number of hyperedges in a friendship r-hypergraph is at least r+1 r n−1 r−1 , and we characterise those hypergraphs which achieve this bound. This generalises a result given by Li and van Rees in the case when r = 3. We also obtain a new upper bound on the number of hyperedges in a friendship r-hypergraph, which improves on a known bound given by Li, van Rees, Seo and Singhi when r = 3.

• Type: Journal
• Year: 2016
• Where: EJC
• Authors: Karen Gunderson, Natasha Morrison, Jason Semeraro
• Keywords: EJC 2016, Information Technology