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 ∆(G) is acyclically edge L-choosable where L = max{2∆(G)− 2,∆(G) + 83}.