There are well-known symmetrical 4- and 6-color arrangements for the dodecahedron, and the rhombic dodecahedron has such arrangements in 3, 4, and 6 colors. What’s different about the (Platonic) dodecahedron that the 3-color arrangement you see here doesn’t make the cut, yet there is one for the rhombic dodecahedron?
The answer: in the other arrangements mentioned, faces of the same color do not share edges. Here, they do, so this one is usually not listed with the others.
Software credit: please visit http://www.software3d.com/Stella.php and try the free download of Stella 4d available there. It’s the program I used to make this image. And, as for the color arrangements mentioned above, they’re pre-loaded into the Stella interface as easy-to-find options.