A Minimum Spanning Tree (MST) of a weighted, connected, undirected graph is unique if and only if there is a unique "best" choice at each step of any MST algorithm (like Kruskal’s or Prim’s). Two commonly known sufficient conditions for MST uniqueness are :

Condition 1 (Distinct edge weights) : If all the edge weights in the graph are distinct (no two edges share the same weight), then the MST is guaranteed to be unique. This is a well-known property.

Condition 2 (Unique minimum-weight edge for each cut): The uniqueness can also be characterized by another condition: for every cut in the graph, the minimum-weight edge that crosses that cut is unique. If there was a tie for the minimum edge across any cut, that could create ambiguity (multiple equally good choices) when forming the MST, potentially leading to multiple MSTs. Ensuring that each cut has a single minimum-weight crossing edge ensures that no such ambiguity exists.

Analyze the given statements :

Statement I : "G has a unique MST if no two edges of G have the same weight."

This statement is true based on standard MST theory. Distinct edge weights guarantee uniqueness of the MST.

Statement II : "G has a unique MST if, for every cut of G, there is a unique minimum-weight edge crossing the cut."

This statement is also true. The uniqueness of the minimum edge in every cut ensures that no alternative MST can be formed by swapping edges of equal weight, thereby guaranteeing a unique MST.

Conclusion :

Both conditions are known criteria for ensuring the uniqueness of an MST. Thus, both I and II are true.

Final Answer : Option C (Both I and II)