TITLE: Suppose there are two different spanning trees for a simple graph. Must they have an edge in common?
QUESTION [5 upvotes]: My instinct is yes, but I don't know how to formalize it into a proof. I still haven't wrapped my head around spanning trees yet. Any thoughts are appreciated!

REPLY [10 votes]: The answer is no. Consider this counterexample:
The red edges form the edges of one spanning tree.
The black edges form the edges of another spanning tree.

REPLY [6 votes]: The complete graph with $5$ verticies provides a counterexample, you can go "around the outside" or "around the star".