Let us call the direct route between adjacent cities a section (of A, B or C type). Given a pair of cities, you can travel between them either clockwise or anticlockwise, but only if the sections you go along are [BC]* in the former case, [BA]* in the latter.
The task is as follows: For every ordered pair of cities (X, Y), construct exactly one travel route (an arc) X->Y provided this is possible, no route if it is impossible. You need to determine the maximum number of routes going along a single section in the solution which minimises this sought value.