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

TL;DR

This paper proves that every planar graph with maximum degree 6 can be colored with at most 20 colors under 2-distance coloring constraints, improving previous bounds.

Contribution

The paper establishes a tighter upper bound of 20 colors for 2-distance coloring of planar graphs with maximum degree 6, advancing prior results.

Findings

Improved upper bound from 21 to 20 colors.

Applicable to all planar graphs with max degree 6.

Enhances understanding of 2-distance coloring limits.

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 $\chi_2(G)$. In this paper, we show that $\chi_2(G) \leq 20$ for every planar graph $G$ with maximum degree at most six, which improves a former bound $\chi_2(G) \leq 21$.