  In this paper an improved bound on the chromatic number of the Pancake graph
$P_n, n\geqslant 2$, is presented. The bound is obtained using a subadditivity
property of the chromatic number of the Pancake graph. We also investigate an
equitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficient
dominating sets is given and optimal equitable $4$-colorings are considered for
small $n$. It is conjectured that the chromatic number of $P_n$ coincides with
its equitable chromatic number for any $n\geqslant 2$.
