  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.
