Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bounds for the Twin-width of Graphs   Bonnet, Kim, Thomass\'{e}, and Watrigant (2020) introduced the twin-width of
a graph. We show that the twin-width of an $n$-vertex graph is less than
$(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph
is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} /
2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at
least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We
also show that the twin-width of the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$
with $1/n\leq p=p(n)\leq 1/2$ is larger than $2p(1-p)n -
(2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for
any positive $\varepsilon$. Lastly, we calculate the twin-width of random
graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the
thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$.

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The categories of this paper are: hep-ph
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The categories of this paper are: supr-con
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: math-ph