Final answer:
The converse of Theorem 15.6 is true; each edge in an MST satisfies the minimum bottleneck property, as exchanging any edge for a smaller one would contradict the definition of an MST.
Step-by-step explanation:
The question asks us to prove the converse of Theorem 15.6, which states that if T is an MST (Minimum Spanning Tree) of a graph with real-valued edge costs, then every edge of T satisfies the minimum bottleneck property. The minimum bottleneck property implies that for any partition of the graph into two non-empty sets that are disconnected, the edge in the MST crossing the cut (if any) is the smallest among all edges that could cross the cut.
To show that every edge of the MST satisfies the minimum bottleneck property, consider an edge e in the MST. If e does not satisfy the property, then there exists another edge e' with a lower cost that also connects the two sets, implying that we could substitute e for e' and get a spanning tree with a lower cost. This contradicts the assumption that T is an MST. Hence, for the tree to be an MST, all its edges must satisfy the minimum bottleneck property.