You're likely not to be the first one to ask this question. Your problem is equivalent to a graph colouring problem, where you have to colour the graph using at most 6 colours, such that no vertex is connected to more than 1 vertex with the same colour as said vertex. There must be literature about that, although most vertex colouring problems have the restriction no edge may connect two nodes of the same colour.

Mathworld has something to say about the number of coloured graphs where no two colours may be adjacent (which is more restrictive than your problem). It suggest that using brute force to calculate the answer to your problem isn't viable.