Saturation Numbers for Berge Cliques

Abstract

Let F be a graph and H be a hypergraph, both embedded on the same vertex set. We say H is a Berge-F if there exists a bijection φ : E(F ) → E(H) such that e ⊆ φ(e) for all e ∈ E(F ). We say H is Berge-F -saturated if H does not contain any Berge-F , but adding any missing edge to H creates a copy of a Berge-F . The saturation number sat k (n, Berge-F ) is the least number of edges in a Berge-F -saturated k-uniform hypergraph on n vertices. We showfor all k, ≥ 3. Furthermore, we provide some sufficient conditions to imply that sat k (n, Berge-F ) = O(n) for general graphs F .