# The 2-Factor Polynomial Detects Even Perfect Matchings

## TL;DR

This paper proves that the 2-factor polynomial for planar trivalent graphs with perfect matchings can count specific 2-factors and detect even perfect matchings, providing a new tool for analyzing graph structures.

## Contribution

The paper introduces a novel application of the 2-factor polynomial to detect even perfect matchings in planar trivalent graphs.

## Key findings

- The 2-factor polynomial counts 2-factors containing a given perfect matching.
- It can distinguish even perfect matchings from others.
- The polynomial serves as an invariant for these graph properties.

## Abstract

In this paper, we prove that the 2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of 2- factors that contain the the perfect matching as a subgraph. Consequently, we show that the polynomial detects even perfect matchings.