On saturation of Berge hypergraphs

Abstract

A hypergraph H = (V (H), E(H)) is a Berge copy of a graph F , if V (F ) ⊂ V (H) and there is a bijection f : E(F ) → E(H) such that for any e ∈ E(F ) we have e ⊂ f (e). A hypergraph is Berge-F -free if it does not contain any Berge copies of F . We address the saturation problem concerning Berge-F -free hypergraphs, i.e., what is the minimum number sat r (n, F ) of hyperedges in an r-uniform Berge-F -free hypergraph H with the property that adding any new hyperedge to H creates a Berge copy of F . We prove that sat r (n, F ) grows linearly in n if F is either complete multipartite or it possesses the following property: ifIn particular, the Berge-saturation number of regular graphs grows linearly in n.