On Wegner's 8-Coloring Theorem for Planar Graphs of Maximum Degree Three

TL;DR

This paper offers a simplified proof for a special case of Wegner's 8-coloring theorem, demonstrating that planar graphs with maximum degree three can be distance-2 colored with at most eight colors, focusing on the complex 5-cycle case.

Contribution

It provides a significantly simplified proof for a specific case of Wegner's conjecture, especially the challenging 5-cycle removal scenario.

Findings

Planar graphs of max degree three are 8-distance-2 colorable.

Simplified proof reduces complexity of Wegner's original argument.

Focus on the 5-cycle case enhances understanding of coloring constraints.

Abstract

We provide a simplified proof of the following special case of Wegner's conjecture: every planar graph of maximum degree at most three admits a distance-2 coloring with at most eight colors. Our main contribution is significant simplification of the most technically challenging part of Wegner's proof: the case involving the removal of a 5-cycle.