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

TL;DR

The work addresses Wegner’s distance-2 coloring problem for planar graphs with maximum degree $3$, establishing an $8$-coloring bound. It introduces the Inside-Outside Lemma to enable a planarity-based decomposition that avoids heavy structural assumptions, and it streamlines the crucial $5$-cycle case via cycle-based reductions. By handling the $4$-cycle and $5$-cycle scenarios under a minimal-counterexample framework, it provides a clearer, more accessible proof that every such graph admits an $8$-coloring. The methods emphasize short-cycle decomposition in planar graphs and may extend to related cycle-removal coloring arguments.

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.

Figures (5)

• Figure 1: Wegner's 7-vertex graph that requires 7 colors in any distance-2 coloring.
• Figure 2:
• Figure 3: A 5-cycle $C$ with interior vertex $u \in G_0$ and exterior vertex $w \in G_1$, both connected to $v \in C$
• Figure 4: 5-cycle with exterior neighbors
• Figure 5:

Theorems & Definitions (4)

• Theorem 1
• Lemma 1: Inside-Outside Lemma
• proof
• Remark 1