Given a graph $G$, we say a $k$-uniform hypergraph $H$ on the same vertex set contains a Berge-$G$ if there exists an injection $\phi:E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each edge $e\in E(G)$. A hypergraph $H$ is Berge-$G$-saturated if $H$ does not contain a Berge-$G$, but adding any edge to $H$ creates a Berge-$G$. The saturation number for Berge-$G$, denoted $\mathrm{sat}_k(n,\text{Berge-}G)$ is the least number of edges in a $k$-uniform hypergraph that is Berge-$G$-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we also prove the existence of nearly-regular $k$-uniform hypergraphs, or $k$-uniform hypergraphs in which every vertex has degree $r$ or $r-1$ for some $r\in \mathbb{Z}$, and less than $k$ vertices have degree $r-1$.
@article{10_37236_8363,
author = {Bethany Austhof and Sean English},
title = {Nearly-regular hypergraphs and saturation of {Berge} stars},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {4},
doi = {10.37236/8363},
zbl = {1431.05106},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8363/}
}
TY - JOUR
AU - Bethany Austhof
AU - Sean English
TI - Nearly-regular hypergraphs and saturation of Berge stars
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/8363/
DO - 10.37236/8363
ID - 10_37236_8363
ER -
%0 Journal Article
%A Bethany Austhof
%A Sean English
%T Nearly-regular hypergraphs and saturation of Berge stars
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/8363/
%R 10.37236/8363
%F 10_37236_8363
Bethany Austhof; Sean English. Nearly-regular hypergraphs and saturation of Berge stars. The electronic journal of combinatorics, Tome 26 (2019) no. 4. doi: 10.37236/8363
