# Every $k$-connected graph is $k$-edge connected.

Give a proof or a counterexample: Every $$k$$-connected graph is $$k$$-edge connected.

Definition: A graph is $$k$$-connected if its connectivity is at least $$k$$. The connectivity of $$G$$ is the minimum size of a vertex $$S$$ such that $$G-S$$ is disconnected or has only one vertex.

Definition: A graph $$G$$ is $$k$$-edge connected if every disconnecting set has at least $$k$$ edges.

I thought this was false, but apparently it is true.

The graph I came up with was:

For instance, it is 2-connected, since we need at least $$2$$ vertices to disconnect it (those vertices in purple on the left). But it is 3-edge connected.

What am I misunderstanding here?

Your example is not a counterexample. It is $2$-edge connected because every disconnecting set has at least $2$ edges. Just because there exists a stricter lower bound of $3$ doesn't mean it isn't $2$-edge connected.