I keep hearing that the subgraphs to the game Instant Insanity must be disjoint. Why is this true? What if the same two colour on the front and back of a cube are the same two colours on the sides of a cube? According to the rules of the game this could still give a solution.
For example in this youtube video it's not clear to me why you need to find a subgraph with no edges in common?
I guess it has to do that if there were the same colour on opposite sides twice than there would be two lines on the graph between those colours. For example if cube 1 has a red front and a green back and a red side and a green other side, then there would be two edges connecting vertices green and red in a graph (and both edges would be labeled 1) so two subgraphs that are disjoint could still be formed. Am I right? Or am I not being clear enough?
Yes, you are exactly correct. If Cube 1 had a red front and a green back and a red side and a green other side, then there would be two edges connecting vertices green and red in a graph (and both edges would be labeled 1) so two subgraphs that are disjoint could still be formed.
The complete graph that one builds for a given set of $N$ cubes always has exactly $3N$ edges. One then selects two disjoint subgraphs of $N$ edges each from these $3N$ total.