Is the following lemma a well known result in graph theory?

I am studying a basic existence result that appears to be simple yet powerful. I have not seen it stated as an important result in graph theory. I have consulted Reinhard Diestel's "Graph Theory" (5th edition, 2017), but could not find it there. So I wanted to ask this question on MO:

**Definition:** Given an $n\times n$ grid with $n^2$ unit squares. If you randomly place exactly 1 diagonal in each unit square, these diagonals (together with the vertices of the grid) form a graph $G$.

**Existence Lemma:** $G$ always contains a path of length $\geq n$.

Above you can see a small example on a $6\times 6$ grid. There is a great graphical example for large $n$ by Joseph O’Rourke https://mathoverflow.net/a/112090/156936

I would be grateful if you could let me know whether this is a well known result, specifically in graph theory.

Is there maybe some more general result from graph theory that implies this particular case? I would be very interested in that.