Square Coloring of Planar Graphs with Maximum Degree at Most Five

TL;DR

This paper proves that the square of any planar graph with maximum degree five can be properly colored with at most 17 colors, improving previous bounds and advancing understanding of graph coloring.

Contribution

It establishes a tighter upper bound of 17 colors for coloring the square of planar graphs with maximum degree five, improving upon the previous bound of 18.

Findings

The square of such graphs is 17-colorable.

Improved upper bound from 18 to 17 colors.

Advances in graph coloring theory for planar graphs.

Abstract

The \textit{square} of a graph $G$, denoted by $G^2$, is obtained from $G$ by adding an edge to connect every pair of vertices with a common neighbor in $G$. In this paper we prove that for every planar graph $G$ with maximum degree at most $5$, $G^2$ admits a proper vertex coloring using at most $17$ colors, which improves the upper bound $18$ recently obtained by Hou, Jin, Miao, and Zhao.