Weak Dynamic Coloring of Planar Graphs

The $k\mathit{\text{-weak-dynamic}}\mathit{\text{number}}$ 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 $\mathrm{m}\mathrm{i}\mathrm{n}\{k,d(v)\}$ 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.