Improved 2-Distance Coloring of Planar Graphs with Maximum Degree 5

TL;DR

This paper proves that any planar graph with maximum degree 5 can be 2-distance colored with at most 17 colors, improving previous bounds and advancing understanding of coloring properties in planar graphs.

Contribution

The paper establishes a tighter upper bound of 17 colors for 2-distance coloring of planar graphs with maximum degree 5, improving the previous bound of 18.

Findings

Proves $oxed{ ext{chi}_2(G) ext{ } extless= 17}$ for all such graphs.

Improves the known upper bound from 18 to 17.

Advances theoretical understanding of 2-distance coloring in planar graphs.

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, denote as $\chi_2(G)$. In this paper, we show that $\chi_2(G) \leq 17$ for every planar graph $G$ with maximum degree $\Delta \leq 5$, which improves a former bound $\chi_2(G) \leq 18$.