Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Separation Dimension and Degree   The "separation dimension" of a graph $G$ is the minimum positive integer $d$
for which there is an embedding of $G$ into $\mathbb{R}^d$, such that every
pair of disjoint edges are separated by some axis-parallel hyperplane. We prove
a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every
graph with maximum degree $\Delta$ has separation dimension less than
$20\Delta$, which is best possible up to a constant factor. We also prove that
graphs with separation dimension 3 have bounded average degree and bounded
chromatic number, partially resolving an open problem by Alon et al. [J. Graph
Theory 2018].

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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: supr-con
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: physics
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: chao-dyn
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The categories of this paper are: econ
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The categories of this paper are: nlin