Weak Dynamic Coloring of 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 $\min \{\mathrm{k},\mathrm{d}(\mathrm{v})\}$ 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.

17 pages, 8 figures