Suppose we have a graph where every edge is colored red or blue. We say that a path is *alternating* if the red and blue edges alternate in it. Our goal is to find many edge/vertex-disjoint alternating paths from a given vertex $s$ to another given vertex $t$. Has this problem been studied before?

Update: It DOES NOT seem to me ANYMORE that the following, Menger-type theorem is true, as pointed out by Ilya in the comments:

Lemma:
We say that a partition of the vertices into $S, T, R, B$ is a *colored cut* if $s\in S, t\in T$, there are no edges between $S$ and $T$ and, except for the edges between $R$ and $B$, the vertices of $R$ are only adjacent to red edges and the vertices of $B$ are only adjacent to blue edges. There are $k$ edge/vertex-disjoint paths from $s$ to $t$ if and only if after deleting any $k-1$ edges/vertices, there is no colored cut.

Update: Gyula Pap told me that if we consider the directed version and edge-disjoint paths, then doubling every vertex (to redin-blueout and bluein-redout) reduces the problem to the monochromatic Menger's theorem from which (if I see well) the above lemma follows. So now I am mainly interested in the vertex-disjoint version.