Hi I am trying to prove the following :
Let G be a connected graph and let T1 and T2 be spanning trees of G. Let e be an edge of T1.
1) Show that T1 \ {e} has exactly two components.
2) Show that there is an edge f of T2 such that (T1 \ {e}) ∪ {f } is connected .
Im not entirely sure on how to answer this , so I was hoping someone could guide me how I can prove these two following statements. Thanks
Let $u, v$ be the two ends of $e$.
Hint for the first one: If $u, v$ are still connected in some way after deleting $e$, was $T_1$ really a tree? If $T_1\setminus \{e\}$ has more than two components, was $T_1$ really connected?
Hint for the second one: There is some path from $u$ to $v$ in $T_2$. Follow that path, and consider the first time in that path that you reach a vertex which isn't reachable from $u$ in $T_1\setminus \{e\}$.