Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nearly-Regular Hypergraphs and Saturation of Berge Stars   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 $r-1$.

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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
7
The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: eess