Saturation Number of Berge Stars in Random Hypergraphs

Abstract

Let $G$ be a graph. We say an $r$-uniform hypergraph $H$ is a Berge-$G$ if there exists a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each $e\in E(G)$. Given a family of $r$-uniform hypergraphs $\mathcal{F}$ and an $r$-uniform hypergraph $H$, a spanning sub-hypergraph $H'$ of $H$ is $\mathcal{F}$-saturated in $H$ if $H'$ is $\mathcal{F}$-free, but adding any edge in $E(H)\backslash E(H')$ to $H'$ creates a copy of some $F\in\mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges in an $\mathcal{F}$-saturated spanning sub-hypergraph of $H$. In this paper, we asymptotically determine the saturation number of Berge stars in random $r$-uniform hypergraphs.

Saturation Numbers for Berge Cliques

“…Saturation numbers for Berge hypergraphs were first studied by the first author and others in [5], where some results on the saturation function for Berge paths, matchings, cycles, and cliques are given. Since the seminal work on saturation for Berge hypergraphs, the topic has gotten significant attention (see [1,2,8,11] for some of the results on the topic). Prior to this work, the following result was the only known result on saturation for Berge cliques.…”