How to solve instant insanity puzzle with graphs.

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So I have a 10 cube insanity puzzle, I constructed the graph (I'll post pc later) based on the colors the problem is that there is just way to many lines and nodes to see the solution. Is there any way to find those paths in the graph without trying every permutation. Guess I was unclear. The puzzle is that each cube has 3 different colors, I need to line them up so that each side will have no repeating colors, so with 10 cubes we have 10 unique colors, now when constructing a graph based on problem I need to find a path that will take me through each node but wont repeat same number path (paths are numbered based on cube numbers). so question is how can I find the path from the graph I attached. enter image description here