Weak Dynamic Coloring of Planar Graphs

TL;DR

This paper proves that all planar graphs can be colored with at most 6 colors to satisfy the 3-weak-dynamic coloring condition, using reducible configurations and list coloring techniques.

Contribution

It introduces a new bound on the 3-weak-dynamic number for planar graphs, improving understanding of their coloring properties.

Findings

All planar graphs have 3-weak-dynamic number at most 6.

Uses reducible configurations and list coloring methods.

Advances coloring theory for planar graphs.

Abstract

The \textit{$k$-weak-dynamic number} of a graph $G$ is the smallest number of colors we need to color the vertices of $G$ in such a way that each vertex $v$ of degree $d(v)$ sees at least $\rm{min}\{k,d(v)\}$ colors on its neighborhood. We use reducible configurations and list coloring of graphs to prove that all planar graphs have 3-weak-dynamic number at most 6.