Cluster expansion formulas and perfect matchings for type B and C

Let $\mathbf{P}_{2n+2}$ be the regular polygon with $2n+2$ vertices, and let $\theta$ be the rotation of 180$^\circ$. Fomin and Zelevinsky proved that $\theta$-invariant triangulations of $\mathbf{P}_{2n+2}$ are in bijection with the clusters of cluster algebras of type $B_n$ or $C_n$. Furthermore, cluster variables correspond to the orbits of the action of $\theta$ on the diagonals of $\mathbf{P}_{2n+2}$. In this paper, we associate a labeled modified snake graph $\mathcal{G}_{ab}$ to each $\theta$-orbit $[a,b]$, and we get the cluster variables of type $B_n$ and $C_n$ which correspond to $[a,b]$ as perfect matching Laurent polynomials of $\mathcal{G}_{ab}$. This extends the work of Musiker for cluster algebras of type B and C to every seed.
Comment: 22 pages. v2: enhanced introduction. arXiv admin note: text overlap with arXiv:2403.11308