An improved linear bound on the number of perfect matchings in cubic graphs

TL;DR

This paper establishes a new lower bound on the number of perfect matchings in cubic bridgeless graphs, improving previous bounds and advancing understanding of their combinatorial structure.

Contribution

It introduces a significantly improved linear lower bound on perfect matchings in cubic bridgeless graphs, surpassing prior constant-based bounds.

Findings

Every cubic bridgeless graph with n vertices has at least 3n/4 - 10 perfect matchings.

This bound differs from the maximal dimension of the perfect matching polytope by more than a constant.

The result advances theoretical understanding of perfect matchings in cubic graphs.

Abstract

We show that every cubic bridgeless graph with n vertices has at least 3n/4-10 perfect matchings. This is the first bound that differs by more than a constant from the maximal dimension of the perfect matching polytope.