Dynamic Coloring and Graph Squares of Regular Graphs

An $r$-dynamic coloring of a graph $G$ is a proper vertex coloring in which every vertex $v$ sees at least $\min\{r,d(v)\}$ distinct colors in its neighborhood. The minimum number of colors in such a coloring is the $r$-dynamic chromatic number $χ_r(G)$. We study dynamic colorings of regular graphs.