2-Distance Coloring of Planar Graphs with Specific Maximum Degree

TL;DR

The paper investigates the 2-distance coloring of planar graphs with maximum degree $\Delta \ge 6$, proving the bound $\chi_2(G) \le 3\Delta + 2$. It adopts a minimal counterexample strategy and defines graphs that are 'proper with respect to' the original graph to enable color extensions, supported by a detailed structural analysis that excludes many problematic local configurations. A discharging argument, guided by Euler's formula, is then crafted through rules $R1$--$R12$ to show all local configurations can be charged nonnegatively, which contradicts the global charge sum of $-8$. Consequently, the conjectured bound holds for all planar graphs with $\Delta \ge 6$, advancing progress toward Wegner's conjecture and tightening the known upper bounds for this regime.

Abstract

A k-distance r-coloring of a graph is a coloring of the vertices of the graph such that if the distance between 2 vertices x and y is less or equal to k, then x and y must have distinct colors. A planar graph is a graph that can be drawn with no edge crossing. We will study the 2-distance coloring of planar graphs with maximum degree at least 6.

Figures (3)

• Figure 1: Illustration of Lemma 2.6
• Figure 2: Illustration of Lemma 2.7
• Figure 3: Illustration of Lemma 2.11

Theorems & Definitions (23)

• Theorem 2.1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5