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A graph is called 1-planar if it can be drawn on the plane so that each edgeis crossed by at most one other edge. In this paper, we establish a localproperty of 1-planar graphs which describes the structure in the neighborhoodof small vertices (i.e. vertices of degree no more than seven). Meanwhile, somenew classes of light graphs in 1-planar graphs with the bounded degree arefound. Therefore, two open problems presented by Fabrici and Madaras [Thestructure of 1-planar graphs, Discrete Mathematics, 307, (2007), 854-865] aresolved. Furthermore, we prove that each 1-planar graph $G$ with maximum degree$\Delta(G)$ is acyclically edge $L$-choosable where$L=\max\{2\Delta(G)-2,\Delta(G)+83\}$.

Structural properties of 1-planar graphs and an application to acyclic edge coloring