Cherry Red wishes to colour all the circles in the diagram (shown below) so that for each circle, there is exactly one circle with the same colour joined to it. What is the smallest number of colours that Cherry needs to complete this task?
When I tried this I got an answer of 4, but I didn’t investigate much further as I wondered whether there is a more technical way (instead of just trying an arrangement) to be sure that you have the smallest possible number. So my question is whether there is a method to be certain you have found the smallest number of colours required?
I can't speak to your broader question, but this graph can be done with two colo(u)rs.