Prove that every - 2-connected graph with n vertices has at least n spanning trees

Question

Please, would someone give me a hint to this proof?

Prove that every 2-connected graph with n-vertices has at least n spanning trees. Does it apply even for 2-edge-connected graph?

Answer 1

Suppose $G$ is a $2$-edge-connected graph on $n$ vertices. Let $T$ be a spanning tree of $G$ with edge set $E(T)=\{e_1,\dots,e_{n-1}\}.$ For each $i\in\{1,\dots,n-1\},$ since $G-e_i$ is connected, we can choose a spanning tree $T_i$ such that $T-e_i\subset T_i\subseteq G-e_i.$ The spanning trees $T,T_1,\dots,T_{n-1}$ are distinct because $E(T)\setminus E(T_i)=\{e_i\}.$

So every $2$-edge connected graph on $n$ vertices has at least $n$ spanning trees. For $n\ge3,$ the cycle $C_n$ is a $2$-connected graph on $n$ vertices with exactly $n$ spanning trees.