Consider the graph $G$. For edges $e_1,e_2 \in E(G),$ let $e_1 \sim e_2$ if either $e_1 = e_2$ or $e_1$ and $e_2$ lie on some common cycle in $G$. I want to be able to prove that $\sim$ is an equivalence relation.
It is apparent to me that by definition, $e \sim e$, and so reflexitivity holds. If $e \sim e'$, the $e'$ must lie in the same cycle as $e$, so $e' \sim e$, and thus symmetry holds. However, I am having some difficulty proving some results related to transitivity. Say $e \sim e' \wedge e' \sim e''$. I want to figure out a cycle that includes all three vertices. If we consider the path $e,e',e'',$ how are we to be sure that there is some cycle that will lead back to $e$? Any assistance would be appreciated.