Cycle of length k with no repeated edges
This problem is NP-Complete.

The $\in NP$ part is simple - guess a trail and check it.

The hardness part comes from the fact Hamiltonicity on a cubic graph is hard, as proved in Garey and Johnson book.

(In a cubic graph, every vertex-simple path walk is edge-simple, as it can not visit a vertex more than once, except maybe the start and end vertex).