  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 $G$, $\chi_d(G)\leq \chi(G)+ c\ln k +1$.
Also, we introduce an upper bound for the dynamic list chromatic number of
regular graphs.
