# Enumerating a type of two-color cycle

I'd be grateful for some pointers to get me started on this.

We have an undirected simple graph $G = (V, E)$. Each edge is colored either blue or red.

I am interested in simple cycles (no repeated vertices or edges other than the starting and ending vertices) that have exactly one blue edge (and the rest of the edges red). Specifically, I want to enumerate these types of cycles in increasing order of length. Do there exist efficient algorithms for this task, or is there a simple reduction to a known problem? What if we restrict $G$ to be planar?

If your graph is $G=(V,E)$ and $B\subseteq E$ are the blue edges, then you can run an appropriate cycle enumeration algorithm on $G-(B\setminus\{ b_i \})$ for each $b_i\in B$. This ensures that at most one blue edge occurs in the graph. Perhaps the selected algorithm can be adapted to start with the edge $b_i$, ensuring that only graphs with exactly one blue edge are enumerated.