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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 subadditivityproperty of the chromatic number of the Pancake graph. We also investigate anequitable coloring of $P_n$. An equitable $(n-1)$-coloring based on efficientdominating sets is given and optimal equitable $4$-colorings are considered forsmall $n$. It is conjectured that the chromatic number of $P_n$ coincides withits equitable chromatic number for any $n\geqslant 2$.

An improved bound on the chromatic number of the Pancake graphs