Cycles

Scenario 1- Allow smaller complexes

If you allow smaller constructs to be built you will need just one bond-edge type and one tile type (T1(Cn) = 1).

Scenario 2 - Allow same size but not smaller complexes

If you do not allow smaller complexes to be built, B2(Cn) = ⌈n/2⌉ and T2(Cn) = ⌈n/2⌉ + 1.

You cannot use fewer tiles since three bond edges of the same type would imply that two edges of the same type are oriented in the same direction around the cycle. The figure below demonstrates why this can result in a smaller structure.

cycle with three edges same orientation
smaller structures resulting from cycle with three edges same orientation

Figure One: Cycle design allowing the creation of smaller complex

Scenario 3 - Allow only larger complexes

The result will be the same as for scenario two, since you will not be able to build any complete complexes of the same size, with tiles of degree two.