A Pfaffian formula for the monomer-dimer model on surface graphs

We consider the monomer-dimer model on weighted graphs embedded in surfaces with boundary, with the restriction that only monomers located on the boundary are allowed. We give a Pfaffian formula for the corresponding partition function, which generalises the one obtained by Giuliani, Jauslin and Lieb for graphs embedded in the disk. Our proof is based on an extension of a bijective method mentioned in their paper, together with the Pfaffian formula for the dimer partition function of Cimasoni-Reshetikhin.