# Example for Berge's theorem on matching

I'm reading about Berge's theorem on matching, but I think I misunderstood the theorem and definitions somehow, because I cannot make sense of the example below. In this example I've marked the matching in pink, and the alternating path in green. This is the only alternating path involving the matching that I could find, and it's clearly not an augmenting path.

I've realized that there is a bigger matching that could be obtained by taking out edge 3 and adding in edges 1 and 2. But since 1325679 is not an alternating path, shouldn't that possibility not be considered by the theorem?

=============================================

Here are the relevant theorem and definitions:

(In fact it doesn't have to contain any edge that is in your matching so far -- say, if you have path graph with four vertices $a-b-c-d$ and start by matching $a$ to $b$, the augmenting path you need to match the other two nodes will consist of the edge $cd$ only).