I need to prove that if a graph $G$ has an edge $e$ contained in two different cycles, there exists a cycle that does not contain $e$.
I drew up an example on a graph, and it's clear to me that the cycles have to intersect at least once, and have at least one common edge ($e$). It occurred to me that then, we could use the other cycle after the point of intersection to return back to the original vertex.
However, what if the two cycles have more vertices in common? Then we would be going through one vertex twice (not a cycle).
So, is there merit in my idea, or how else would one prove that?