Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On an extremal problem in the class of 1-planar graphs   A graph $G=(V,E)$ is called 1-planar if it admits a drawing in the plane such
that each edge is crossed at most once. In this paper, we study bipartite
$1$-planar graphs with prescribed numbers of vertices in partite sets.
Bipartite 1-planar graphs are known to have at most $3n-8$ edges, where $n$
denotes the order of a graph. We show that maximal-size bipartite $1$-planar
graphs which are almost balanced have not significantly fewer edges than
indicated by this upper bound, while the same is not true for unbalanced ones.
We prove that maximal possible sizes of bipartite $1$-planar graphs whose one
partite set is much smaller than the other one tends towards $2n$ rather than
$3n$. In particular, we prove that if the size of the smaller partite set is
sublinear in $n$, then $|E|=(2+o(1))n$, while the same is not true otherwise.

19
The categories of this paper are: eess
25
The categories of this paper are: patt-sol
37
The categories of this paper are: supr-con
6
The categories of this paper are: astro-ph
30
The categories of this paper are: q-alg
20
The categories of this paper are: acc-phys
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex