Complete Bipartite Graphs
- Scenario 1
T1(Km,n) =
{
1 if m = n and both are even,
2 otherwise.
- Scenario 2
-Proposition. T2(Km,n) =
{
2 if gcd(m,n) = 1
3 if gcd(m,n) > 1
-Proposition. B2(Km,n) =
{
1 if gcd(m,n) = 1
2 if gcd(m,n) > 1
- Scenario 3
- Observation. Suppose we have a complete complex for Km,n from whose tiles no smaller or same size complexes may be formed. If we think of the edges as being oriented from unhatted to hatted cohesive ends, then all edges of a given bond-edge type must be oriented from X to Y or vice versa. This follows because two oppositely oriented edges of the same bond-edge type may exchange
half-edges to form a nonisomorphic same size complete complex. Thus, we may assume that all edges are oriented from X to Y . //
- Observation. Suppose we have a complete complex for Km,n from whose tiles no smaller or same size complexes may be formed, with the edges oriented from X to Y . Then there can be no two non-adjacent edges with the same bond-edge type, or they may exchange half-edges to form a same size complete complex.
- Proposition. T3 (Km,n) = m + 1.
- Proposition. B3 (Km,n) = m.