Cluster expansion formulas and perfect matchings for type B and C

Let P₂ₙ₊₂ be the regular polygon with $2n+2$ vertices, and let θ be the rotation of 180^∘. Fomin and Zelevinsky showed that θ-invariant triangulations of P₂ₙ₊₂ are in bijection with the clusters of cluster algebras of type Bₙ or Cₙ. Moreover, cluster variables correspond to the orbits of the action of θ on the diagonals of P₂ₙ₊₂. In this paper, we associate a labeled modified snake graph Gab to each θ-orbit $[a,b]$, and we get the cluster variables of type Bₙ and Cₙ which correspond to $[a,b]$ as perfect matching Laurent polynomials of Gab. This extends the work of Musiker for cluster algebras of type B and C to every seed.