Each cycle in the graph, the edge with the minimum weight belongs to MST
Following is a simple counterexample: Take a complete graph on four vertices: ${u_{1},u_{2},u_{3},u_{4}}$, with edge weights $w(u_{1},u_{2}) = 1$, $w(u_{2},u_{3}) = 2$, $w(u_{1},u_{3}) = 4$, $w(u_{1},u_{4}) = 5$, $w(u_{2},u_{4}) = 3$, and $w(u_{3},u_{4}) = 6$. Here MST is composed of edges: $(u_{1},u_{2})$, $(u_{2},u_{3})$, and $(u_{2},u_{4})$ with total weight $6$. And, no edge of the cycle $(u_{1},u_{3},u_{4})$ is part of the MST.