Define a valid "coloring" to occur when no two faces with a common edge share the same color. Given two colors, there is a single way to color an octahedron
(Ball and Coxeter 1987, pp. 238-239). Given three colors, there is one way to
color a cube (Ball and Coxeter 1987, pp. 238-239) and
144 ways to color an icosahedron (Ball and Coxeter
1987, pp. 239-242). Given four colors, there are two distinct ways to color
a tetrahedron (Ball and Coxeter 1987, p. 238)
and four ways to color a dodecahedron, consisting
of two enantiomorphous ways (Steinhaus 1999, pp. 196-198; Ball and Coxeter 1987,
p. 238). Given five colors, there are four ways to color an icosahedron.
Given six colors, there are 30 ways to color a cube (Steinhaus
1999, p. 167). These values are related to the chromatic
polynomial of the corresponding dual skeleton graph, which however overcounts
since it does not take rotational equivalence of colorings in the original solid
into account.

The following table gives the numbers of ways to color faces of various solids using at most
colors (with no restriction about colors on adjacent faces). This can be computed
by finding the graph automorphisms of the skeleton of
the polyhedron, removing the symmetries that invert a face (leaving pure rotational
symmetries only), then finding the induced symmetry group for the faces and applying
the Pólya enumeration theorem.