Weak Dynamic Coloring of Planar Graphs

The 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 min{k,d(v)}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{k,d(v)\}$$\end{document} 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.