Edge connectivity of a graph given the number of edge disjoint paths

[Superyoshiom]

What can we infer about the connectivity and edge connectivity of a graph given the number of edge disjoint paths?

So the number of edge disjoint u,v paths in a graph is x. Doing this problem, I thought back to Menger's theorem, and thought that the graph is x-edge-connceted and so the number of cut sets between any two vertices is also x. However I'm not sure how to find find out anything about the vertex connectivity since all I was given was the number of edge disjoint paths.

 

[haruspex]

number of cut sets

Do you mean minimal size of cut sets?

However I'm not sure how to find find out anything about the vertex connectivity since all I was given was the number of edge disjoint paths.

Perhaps it doesn’t tell you much about connectivity. What extreme examples can you construct?