Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Minimizing the number of edges in $K_{s,t}$-saturated bipartite graphs   This paper considers an edge minimization problem in saturated bipartite
graphs. An $n$ by $n$ bipartite graph $G$ is $H$-saturated if $G$ does not
contain a subgraph isomorphic to $H$ but adding any missing edge to $G$ creates
a copy of $H$. More than half a century ago, Wessel and Bollob\'as
independently solved the problem of minimizing the number of edges in
$K_{(s,t)}$-saturated graphs, where $K_{(s,t)}$ is the `ordered' complete
bipartite graph with $s$ vertices from the first color class and $t$ from the
second. However, the very natural `unordered' analogue of this problem was
considered only half a decade ago by Moshkovitz and Shapira. When $s=t$, it can
be easily checked that the unordered variant is exactly the same as the ordered
case. Later, Gan, Kor\'andi, and Sudakov gave an asymptotically tight bound on
the minimum number of edges in $K_{s,t}$-saturated $n$ by $n$ bipartite graphs,
which is only smaller than the conjecture of Moshkovitz and Shapira by an
additive constant. In this paper, we confirm their conjecture for $s=t-1$ with
the classification of the extremal graphs. We also improve the estimates of
Gan, Kor\'andi, and Sudakov for general $s$ and $t$, and for all sufficiently
large $n$.

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The categories of this paper are: nucl-ex
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The categories of this paper are: mtrl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: acc-phys