I am reading a lecture about dimers by R.Kenyon (http://arxiv.org/pdf/0910.3129v1.pdf).
I have a question concerned to the honeycomb graph $H_n$ embedded on a torus (see the picture at page 24 of the above link).
Let $M_0$ be the dimer configuration (or by other words, perfect matching) containing all the horizontal edges of $H_n$. Take another dimer configuration $M$ on $H_n$, then the symmetric difference $M\Delta M_0$ consists of loops (or by other words, composition cycles). By lemma 5 (page 25) of the lecture, the author claims that the number of loops is $GCD(h_x,h_y)$, where $h_x=N_b/n$, $h_y=N_c/n$ with $N_b, N_c$ the number of b-type edges and c-type edges respectively of $M$.
I don't understand this statement. Someone can help me? (Consequently, this implies that all the loops of $M\Delta M_0$ have a same length, it's an interseting result!)