The number of perfect matchings in a brick

TL;DR

This paper disproves a conjecture by constructing an infinite family of bricks with many perfect matchings, showing that the minimum number of perfect matchings in bricks can be significantly larger than previously conjectured.

Contribution

The paper introduces a new family of bricks with many perfect matchings, providing a counterexample to a longstanding conjecture about the minimum number of perfect matchings.

Findings

Constructed an infinite family of bricks with many perfect matchings

Showed that for even n > 17, bricks can have at least 0.625n perfect matchings

Disproved the conjecture that bricks always have at least n-1 perfect matchings

Abstract

A 3-connected graph is a brick if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching decomposition procedure of Kotzig, and Lovasz and Plummer. Lucchesi and Murty conjectured that there exists a positive integer N such that for every n>N, every brick on n vertices has at least n-1 perfect matchings. We present an infinite family of bricks such that for each even integer n (n > 17), there exists a brick with n vertices in this family that contains [0:625n] perfect matchings, showing that this conjecture fails.