Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tight Upper Bounds on the Crossing Number in a Minor-Closed Class   The crossing number of a graph is the minimum number of crossings in a
drawing of the graph in the plane. Our main result is that every graph $G$ that
does not contain a fixed graph as a minor has crossing number $O(\Delta n)$,
where $G$ has $n$ vertices and maximum degree $\Delta$. This dependence on $n$
and $\Delta$ is best possible. This result answers an open question of Wood and
Telle [New York J. Mathematics, 2007], who proved the best previous bound of
$O(\Delta^2 n)$. We also study the convex and rectilinear crossing numbers, and
prove an $O(\Delta n)$ bound for the convex crossing number of bounded
pathwidth graphs, and a $\sum_v\deg(v)^2$ bound for the rectilinear crossing
number of $K_{3,3}$-minor-free graphs.

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The categories of this paper are: mtrl-th
14
The categories of this paper are: math-ph
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
26
The categories of this paper are: bayes-an
28
The categories of this paper are: alg-geom
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc