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

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

2010-08-30

arxiv.org/abs/1008.5000v2

Scientia Sinica Mathematica, 2010, 40, 1025--1032

Computer Science

Discrete Mathematics

Please cite this published article as: X. Zhang, G. Liu, J.-L. Wu. Structural properties of 1-planar graphs and an application

Scientific paper

10.1360/012009-678

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 $L=\max\{2\Delta(G)-2,\Delta(G)+83\}$.

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