I just encountered a question of the MST:
Assume $G$ is a weighted connected graph, and there is a cycle $C$ of length $m$, and the weights of these $m$ edges are all equal to the minimum value of edge weights in $E(G)$.[ $E(G)$ is the set of the edges. ]
Prove: there are at least $m$ different minimum spanning trees of $G$.
My solution is:
First prove that the minimum spanning tree must contain one of these $m$ least weight edges.
Then assume $T$ contains edge $e_i, i \in \{1,2,\dots, m\}$. Add another edge $e_j, j \in \{1,2,\dots, m\}, j\neq i$ to the minimum spanning tree $T$.
But I found it difficult in step 2... It seems useless in proving there are at least $m$ different minimum spanning trees.