Let $G$ be a connected graph. Prove that if $E(G)$ is greater than or equal to $V(G)$ then $G$ contains at least three edges that are not bridges.

Question

I was wondering how to solve this problem.

The idea is something along these lines if there are more edges than vertices (or equal), you have a cycle.

Since this is only possible with Graphs of order $3$ or more, we have a cycle of $3$ or more.

This means that you could take any of those edges out, and the graph is connected.

Making it not a bridge. Am I right?

Answer 1

As stated when we have an undirected graph of more than/equal to edges, there exist cycles since you have to have an extra connection somewhere. This can be removed, making it not a bridge.