Saturation of Berge Hypergraphs

TL;DR

This paper investigates the minimum size of hypergraphs that are saturated with respect to Berge-$F$ structures, focusing on various graphs like triangles, paths, and cycles within the context of $k$-uniform hypergraphs.

Contribution

It introduces and analyzes the saturation numbers for Berge-$F$ structures in $k$-uniform hypergraphs, extending classical saturation concepts to hypergraph settings.

Findings

Determined saturation numbers for Berge triangles and paths.

Established bounds for Berge cycles and stars.

Extended classical graph saturation results to hypergraphs.

Abstract

Given a graph $F$, a hypergraph is a Berge-$F$ if it can be obtained by expanding each edge in $F$ to a hyperedge containing it. A hypergraph $H$ is Berge-$F$-saturated if $H$ does not contain a subgraph that is a Berge-$F$, but for any edge $e\in E(\overline{H})$, $H+e$ does. The $k$-uniform saturation number of Berge-$F$ is the minimum number of edges in a $k$-uniform Berge-$F$-saturated hypergraph on $n$ vertices. For $k=2$ this definition coincides with the classical definition of saturation for graphs. In this paper we study the saturation numbers for Berge triangles, paths, cycles, stars and matchings in $k$-uniform hypergraphs.