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

TL;DR

This paper explores the local structure of 1-planar graphs, introduces new classes of light graphs, solves open problems, and proves acyclic edge list coloring bounds for these graphs.

Contribution

It establishes local properties of 1-planar graphs, introduces new light graph classes, and proves bounds for acyclic edge coloring.

Findings

Solved two open problems on 1-planar graphs.

Identified new classes of light graphs with bounded degree.

Proved acyclic edge L-choosability bounds for 1-planar graphs.

Abstract

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