I want to create graphs where each vertex has three edges, and is connected by these three edges to three distinct vertices. I'd like to know the number of vertices for which this is possible. By playing around a bit, I've found that it's possible for graphs with 4, 8, and 12 vertices. If v is the number of vertices, it's easy to see that a necessary (but not sufficient) condition is that [tex]3v/2 \equiv 0 (mod 3) [/tex]. The attached image shows exactly what I'm looking for. The graphs for the tetrahedron, cube, and dodecahedron all satisfy my criteria, while the others do not. Thanks in advance!