# True false question related to graph having a unique Minimum weight spanning tree

• You have an undirected graph $G$
• $G$ has a cycle in it
• That cycle has an edge $e$
• e is a unique lightest weight edge in that cycle

Is it true that $e$ is part of every Minimum weight spanning tree of $G$?

I cant find any example to disprove this statement so I am assuming it is true. Do you think the same? or can you find a counter example?

• Does $G$ have only that one cycle, or can it have others? – Daniel Fischer Oct 8 '14 at 22:15
• @DanielFischer it can have other cycles also – Krimson Oct 8 '14 at 22:16

ASCII art:

    A---1---B
|\     /|
| 1   1 |
|  \ /  |
1   C   1
|  / \  |
| 2   3 |
|/     \|
D---4---E


Consider the cycle $CDE$.