Let $G = (V, E)$ and $G' = (V', E')$ be two graphs, and let $f: V \rightarrow V'$ be a surjection. Then $f$ is a covering map from $G$ to $G'$ if for each $v \in G$, the restriction of $f$ to the neighbourhood of $v$ is a bijection onto the neighbourhood of $f(v) \in V'$ in $G'$.
My question (homework) is how to easily prove that if there exists a cycle in $G$, there also exists a cycle in $G'$?
I have a proof based on the size of the preimage of each vertex of $G'$. But, it seems to complicate. I would like to know your point of view.
Thanks a lot in advance.