I know Prof. Loh mentioned that straight multiplication doesn't work, but I think this way of multiplication might?
For the first vertex, there are 3 choices for what color it might be.
For the second vertex, there are 2 choices- the choices being the colors that were not the one selected for the first vertex.
For the third vertex, there is supposed to be 2 choices, but because we're concerned about whether its color is the same as that of the first vertex, we can split it into 2 cases: one in which they're the same (1 choice of color for that), and the other in which the colors are different (1 choice of color for that).
If vertex 1 and vertex 3 have the same color, then there are 2 choices for the color of vertex 4.
If the colors are different, there is only 1 choice for the color of vertex 4.
Therefore, the answer is 3x2x(1+2)=18.
Actually, never mind. I just realized that Prof. Loh just explained it in the next part.