Graphical Condensation Generalizations Involving Pfaffians and Determinants

TL;DR

This paper extends graphical condensation techniques, which relate perfect matchings in plane graphs, by generalizing identities and expressing matchings as Pfaffians or determinants involving subgraph matchings.

Contribution

It introduces new generalizations of graphical condensation identities and formulates the number of perfect matchings as Pfaffians or determinants involving subgraph matchings.

Findings

Generalized identities for nonbipartite graphs

Expressed perfect matchings as Pfaffians and determinants

Enhanced combinatorial enumeration methods

Abstract

Graphical condensation is a technique used to prove combinatorial identities among numbers of perfect matchings of plane graphs. Propp and Kuo first applied this technique to prove identities for bipartite graphs. Yan, Yeh, and Zhang later applied graphical condensation to nonbipartite graphs to prove more complex identities. Here we generalize some of the identities of Yan, Yeh, and Zhang. We also describe the latest generalization of graphical condensation in which the number of perfect matchings of a plane graph is expressed as a Pfaffian or a determinant where the entries are also numbers of perfect matchings of subgraphs.