Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Structural properties of 1-planar graphs and an application to acyclic
  edge coloring   A graph is called 1-planar if it can be drawn on the plane so that each edge
is crossed by at most one other edge. In this paper, we establish a local
property of 1-planar graphs which describes the structure in the neighborhood
of small vertices (i.e. vertices of degree no more than seven). Meanwhile, some
new classes of light graphs in 1-planar graphs with the bounded degree are
found. Therefore, two open problems presented by Fabrici and Madaras [The
structure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] are
solved. Furthermore, we prove that each 1-planar graph $G$ with maximum degree
$\Delta(G)$ is acyclically edge $L$-choosable where
$L=\max\{2\Delta(G)-2,\Delta(G)+83\}$.

36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
19
The categories of this paper are: eess
12
The categories of this paper are: nucl-th