Let $G$ be a connected graph. If $G$ has no cut vertices, then G has no bridges.

Question

Prove, disprove, or give a counterexample: Let $G$ be a connected graph. If $G$ has no cut vertices, then $G$ has no bridges.

Answer 1

Prove that if $G$ has a bridge, then $G$ has a cut vertex. Specifically, prove the an endpoint of a bridge is a cut vertex. This statement is logically equivalent to the original problem (http://en.wikipedia.org/wiki/Contraposition).

Edit: You cannot actually prove the above statement for 2 vertices because there is a rather simple counter example as explained in the comments. But for larger graphs you can ensure that one of the vertices are a cut vertex as long as it is part of a connected component of size 3 or more.

Answer 2

Assuming at least $3$ vertices, this is true.

With $2$ vertices, just think of the only connected graph possible.

See @Danikar's good answer.