Graph spanning tree proof

Question

Let $C^*$ be a set of edges of a connected graph $G$. Show that if $C^*$ has an edge in common with each spanning tree of $G$, then $C^*$ contains a cutset.

Obtain a corresponding result for cycles.

How could I begin this proof?

Answer 1

Each edge in a spanning tree is a cut edge. That is, if $e$ is a cut-edge of $T$, then $T - \{e\}$ leaves $T$ disconnected. A cut set $K$ is a set of edges such that $E(G) - K$ will leave $G$ disconnected. So what happens if we remove an edge from each spanning tree of $G$? Won't that leave $G$ disconnected?