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

## TL;DR

This paper derives a Pfaffian formula for the monomer-dimer model's partition function on surface-embedded graphs with boundary, extending previous disk-specific results to more general surfaces.

## Contribution

It introduces a generalized Pfaffian formula for the monomer-dimer model on surface graphs, expanding the applicability of existing formulas beyond disk embeddings.

## Key findings

- Provides a Pfaffian formula for the monomer-dimer partition function on surface graphs.
- Extends previous disk-specific formulas to more general surface embeddings.
- Uses an extended bijective method combined with Cimasoni-Reshetikhin's Pfaffian formula.

## Abstract

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.