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2-distance 20-coloring of planar graphs with maximum degree 6

Kengo Aoki

Abstract

A 2-distance $k$-coloring of a graph $G$ is a proper $k$-coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of $G$ is the minimum $k$ such that $G$ has a 2-distance $k$-coloring, denoted by $χ_2(G)$. In this paper, we show that $χ_2(G) \leq 20$ for every planar graph $G$ with maximum degree at most six, which improves a former bound $χ_2(G) \leq 21$.

2-distance 20-coloring of planar graphs with maximum degree 6

Abstract

A 2-distance -coloring of a graph is a proper -coloring such that any two vertices at distance two or less get different colors. The 2-distance chromatic number of is the minimum such that has a 2-distance -coloring, denoted by . In this paper, we show that for every planar graph with maximum degree at most six, which improves a former bound .

Paper Structure

This paper contains 3 sections, 16 theorems, 2 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Reducible configurations
  3. Discharging

Key Result

Theorem 1.2

If $G$ is a planar graph with maximum degree $\Delta \leq 6$, then $\chi_2(G) \leq 20$.

Figures (1)

  • Figure 1: Illustrations of Lemma \ref{['lem9']}(4.3).

Theorems & Definitions (32)

  • Conjecture 1.1
  • Theorem 1.2
  • Remark
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 22 more