Cluster expansion formulas and perfect matchings for type B and C

Let P₂₍+₂ be the regular polygon with 2n+2 vertices, and let be the rotation of 180^. Fomin and Zelevinsky showed that -invariant triangulations of P₂₍+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 on the diagonals of P₂₍+₂. In this paper, we associate a labeled modified snake graph G₀₁ to each -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 G₀₁. This extends the work of Musiker for cluster algebras of type B and C to every seed.