# Perfect matchings and derangements on graphs

## TL;DR

This paper proves that in bipartite graphs, each perfect matching intersects at least half of all perfect matchings, with implications for permanents, derangements, and permutations on graphs.

## Contribution

It establishes a new lower bound on the intersection size of perfect matchings in bipartite graphs, connecting combinatorial and algebraic graph properties.

## Key findings

- Each perfect matching intersects at least half of all perfect matchings in a bipartite graph.
- Equivalent formulations relate to the permanent of the adjacency matrix and derangements on graphs.
- Provides related results and open questions in the study of graph matchings.

## Abstract

We show that each perfect matching in a bipartite graph $G$ intersects at least half of the perfect matchings in $G$. This result has equivalent formulations in terms of the permanent of the adjacency matrix of a graph, and in terms of derangements and permutations on graphs. We give several related results and open questions.