I was asked to show that there is no solution to the following four cubes instant insanity problem

The game Instant Insanity is played with four cubes. Each face of a cube is colored with one of four colors. The object of the game is to stack the cubes one on top of another so that on each long face of the resulting shape, all four colors show up exactly once. Note that there are four such faces, which we will describe as top, bottom, front, and back. The picture above shows the four cubes placed together in a way that you can see the top and front faces.

One way is to draw out all the possible combinations of the two subgraphs and show that non of the pairs could simultaneously satisfy the requirements of the solution.

Are there simpler ways of proving a Instant Insanity problem has no solution?

I guess the key is the hamiltonicity of the subraphs...In particular the subgraph have to be connected so you cannot choose edges with loops because the condition "*Each vertex must have degree 2*" would force you to disconnect your graph. It reduces a lot the possibilies when you try to construct the Hamiltonian subraphs.