Complete Graphs

Scenario 1 - allow smaller complexes

T₁(K_n) = 1 if n is odd, and 2 if n is even.

If G is a bipartite graph with all vertices of the same degree in the same bipartition, then T1(G) = av (G).

Scenario 2

Proposition. T₂(K_n) = 2 if n is even, and 3 if n is odd.

Proposition. B₂(K_n) = 1 if n is even, and 2 if n is odd.

Scenario 3

Corollary. T₃(K_n) = n.

Proposition. B₃(K_n) = n − 1.