Could somebody please help me with this problem? I realise it's pretty simple, but for some reason can't really wrap my head around it.
Determine if these are true or false and prove it:
(i) If $e$ is an edge with the least cost in a cycle of $G$, then $e$ belongs to a minimum spanning tree of $G$.
(ii) If $e$ is an edge of a minimum spanning tree of $G$, then $e$ is an edge with the least cost in some cutset in $G$.
Thank you!
(i) Let $G \cong K_4$, $V(G) = \{\, 1, 2, 3, 4\,\}$ and $w\colon V(G) \to \mathbb{R}$ be the following weight function: $w(\{\,1, 2\,\}) = w(\{\,1, 3\,\}) = w(\{\,1, 4\,\}) = 1$, $w(\{\,2, 3\,\}) = w(\{\,2, 4\,\}) = 3$, $w(\{\,3, 4\,\}) = 2$. Then $\{\,3, 4\,\}$ is an edge with the least weight in a cycle $2, 3, 4, 2$ of $G$. However it obviously doesn't belong to the only minimum spanning tree.
(ii) Let's remove edge $e$ of the minimum spanning tree and break the tree into two components with sets $U$ and $V$ of vertices. Now we can consider the cut set between these two sets of vertices. If there is an edge $e'$ such that $w(e') < w(e)$ then we can remove $e$ from spanning tree and add edge $e'$ to get a spanning tree of lower weight. So if $e$ belongs to minimum spanning tree then it has the minimum weight in some cutset.