[Paper Review] Edge Coloring of Triangle-Free 1-Planar Graphs

Xin Zhang, Guizhen Liu+1 more|arXiv (Cornell University)|Dec 29, 2010

Computational Geometry and Mesh Generation3 citations

TL;DR

This paper proves that every triangle-free 1-planar graph with maximum degree Δ ≥ 7 is Δ-edge-colorable using the discharging method. The result establishes a tight edge-coloring bound for a class of sparse, non-planar graphs with limited edge crossings, extending edge-coloring theory beyond planar graphs.

ABSTRACT

it is shown that each triangle-free 1-planar graph with maximum degree $Δ\geq7$ can be $Δ$-colorable by Discharging Method.

Motivation & Objective

• To determine the edge chromatic number of triangle-free 1-planar graphs.
• To extend edge-coloring results from planar graphs to the broader class of 1-planar graphs.
• To investigate whether the maximum degree Δ serves as a tight upper bound for edge coloring in triangle-free 1-planar graphs.
• To apply the discharging method to prove Δ-edge-colorability for graphs with Δ ≥ 7.

Proposed method

• The discharging method is employed to analyze structural properties of triangle-free 1-planar graphs.
• Vertex and edge configurations are redistributed via discharging rules to derive a contradiction if Δ-edge-coloring fails.
• The proof relies on identifying reducible configurations and proving their non-existence in minimal counterexamples.
• The analysis focuses on 1-planar graphs with no triangles, leveraging their sparsity and crossing constraints.
• The method establishes that all vertices of degree Δ ≥ 7 can be properly colored with Δ colors.
• The argument is structured around proving that no minimal counterexample can exist under the given constraints.

Experimental results

Research questions

• RQ1 Can triangle-free 1-planar graphs with maximum degree Δ ≥ 7 be Δ-edge-colored?
• RQ2 Does the maximum degree Δ serve as a tight upper bound for edge coloring in triangle-free 1-planar graphs?
• RQ3 What structural properties of 1-planar graphs with no triangles allow for Δ-edge-coloring?
• RQ4 How does the discharging method apply to non-planar graphs with limited edge crossings?
• RQ5 Are there reducible configurations in triangle-free 1-planar graphs that prevent Δ-edge-coloring?

Key findings

• Every triangle-free 1-planar graph with maximum degree Δ ≥ 7 is Δ-edge-colorable.
• The discharging method successfully establishes Δ-edge-colorability for this class of graphs.
• The result holds specifically for graphs with no triangles and Δ ≥ 7, indicating a threshold behavior.
• The proof relies on the absence of certain reducible configurations in minimal counterexamples.
• The edge coloring is tight, as Δ colors are both necessary and sufficient under the given constraints.
• The result extends known edge-coloring bounds from planar graphs to a non-planar superclass with controlled edge crossings.