A Novel Approach to Counting Perfect Matchings of Graphs

TL;DR

This paper introduces a new theoretical framework inspired by surface cohomology to count perfect matchings in graphs, providing novel proofs and insights into classical combinatorial objects.

Contribution

It presents a novel approach based on relative cohomology to count perfect matchings and rederives the Aztec Diamond theorem with connections to alternating sign matrices.

Findings

Reproves the Aztec Diamond theorem using the new framework

Establishes a link between perfect matchings and surface cohomology

Shows how alternating sign matrices emerge naturally in this context

Abstract

We build a new perspective to count perfect matchings of a given graph. This idea is motivated by a construction on the relative cohomology group of surfaces. As an application of our theory, we reprove the celebrated Aztec Diamond theorem, and show how alternating sign matrices naturally arises through this framework.