Is the Cut-Property Two way?

Question

According to the cut-property of an MST, if an edge belongs to the cut set of a graph, then the MST contains this edge.

However, is it true that if an MST contains an edge, then this edge must belong to the cut set?

Answer 1

You did not reproduce the cut property correctly. The cut property is (source: Wikipedia

For any cut C of the graph, if the weight of an edge e in the cut-set of C is strictly smaller than the weights of all other edges of the cut-set of C, then this edge belongs to all MSTs of the graph.

So, it is not enough for an edge to belong to a cut set of any cut. It must furthermore be the unique minimum-weight edge in that cut set.

So, what about the inversion: If an edge belongs to the MST, then there must be a cut whose cut set contains this edge.

This is obviously true as you can define arbitrary cuts. Including one that includes the edge in its cut set.

What about the more precise formulation: If an edge belongs to the MST, then there must be a cut whose cut set contains this edge and where the edge has strictly smaller weight than all other edges of the cut set.

This is not true. Just consider a graph where all edges are equal. Then there is no edge that fulfills the criterion (no edge has strictly smaller weight than any other), yet the MST is not empty.