  A {\it dynamic $k$-coloring} of a graph $G$ is a proper $k$-coloring of the
vertices of $G$ such that every vertex of degree at least 2 in $G$ will be
adjacent to vertices with at least 2 different colors. The smallest number $k$
for which a graph $G$ can have a dynamic $k$-coloring is the {\it dynamic
chromatic number}, denoted by $\chi_d(G)$. In this paper, we investigate the
dynamic 3-colorings of claw-free graphs. First, we prove that it is
$NP$-complete to determine if a claw-free graph with maximum degree 3 is
dynamically 3-colorable. Second, by forbidding a kind of subgraphs, we find a
reasonable subclass of claw-free graphs with maximum degree 3, for which the
dynamically 3-colorable problem can be solved in linear time. Third, we give a
linear time algorithm to recognize this subclass of graphs, and a linear time
algorithm to determine whether it is dynamically 3-colorable. We also give a
linear time algorithm to color the graphs in the subclass by 3 colors.
