Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Three classes of 1-planar graphs   A graph is called 1-planar if it can be drawn in the plane so that each of
its edges is crossed by at most one other edge. In this paper we decompose the
set of all 1-planar graphs into three classes $\mathcal C_0, \mathcal C_1$ and
$\mathcal C_2$ with respect to the types of crossings and present the
decomposition of 1-planar join products.
  Zhang \cite{z} proved that every $n$-vertex 1-planar graph of class $\mathcal
C_1$ has at most $\frac{18}{5}n-\frac{36}{5}$ edges and a $\mathcal
C_1$-drawing with at most $\frac 35 n-\frac 65$ crossings. We improve these
results. We show that every $\mathcal C_1$-drawing of a 1-planar graph has at
most $\frac 35 n-\frac 65$ crossings. Consequently, every $n$-vertex 1-planar
graph of class $\mathcal C_1$ has at most $\frac{18}{5}n-\frac{36}{5}$ edges.
Moreover, we prove that this bound is sharp.

30
The categories of this paper are: q-alg
28
The categories of this paper are: alg-geom
8
The categories of this paper are: hep-ex
6
The categories of this paper are: astro-ph
12
The categories of this paper are: nucl-th
20
The categories of this paper are: acc-phys
18
The categories of this paper are: econ
7
The categories of this paper are: hep-th
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
21
The categories of this paper are: adap-org
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess