On Dynamic Coloring of Graphs

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. In this paper we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that there is a constant $c$ such that for every $k$-regular graph $G$, ${\chi }_d(G)\le \chi (G)+c\ln k+1$. Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs.