A $2^{O(k)}n$ algorithm for $k$-cycle in minor-closed graph families

Let $\mathcal{C}$ be a proper minor-closed family of graphs. We present a randomized algorithm that given a graph $G\in \mathcal{C}$ with $n$ vertices, finds a simple cycle of size $k$ in $G$ (if exists) in $2^{O(k)}n$ time. The algorithm applies to both directed and undirected graphs. In previous linear time algorithms for this problem, the runtime dependence on $k$ is super-exponential. The algorithm can be derandomized yielding a $2^{O(k)}n\log n$ time algorithm.