# Determining whether a graph is bipartite or not

I would like to decide whether the following graph is bipartite or not:

One way to do this is to determine its chromatic number first, which is obviously $2$, and since every graph with chromatic number $2$ is bipartite, we'd have our answer.

Another way to do this would be to use the fact that every cycle in a bipartite graph has even length. I don't see why this should be true in this case though, so I might have a misunderstanding about what a cycle actually is.

For example, when I start at the top left of the outer square, then I go into the inner square and walk around it, then I leave it at the bottom left and close the cycle by walking to the top left vertex again. But in this case, the cycle would consist of $7$ vertices, because I have to count the vertex where I started twice (since it's the same vertex where I end). So, I would have found an odd cycle here, and this contradicts my observations from before.

What is wrong here?