The maximum number of perfect matchings in graphs with a given degree sequence

TL;DR

This paper establishes an upper bound on the number of perfect matchings in simple graphs with a specified degree sequence, identifying the extremal graphs where this bound is achieved.

Contribution

It provides a sharp upper bound for perfect matchings based on degree sequences and characterizes the extremal graphs attaining this bound.

Findings

Maximum number of perfect matchings is bounded by a product involving degree factorials.

The bound is tight and achieved by unions of complete balanced bipartite graphs.

The result generalizes previous bounds and characterizes extremal structures.

Abstract

We show that the number of perfect matching in a simple graph $G$ with an even number of vertices and degree sequence $d_1,d_2, ..., d_n$ is at most $\prod_{i=1}^n (d_i !)^{\frac{1}{2d_i}}$. This bound is sharp if and only if $G$ is a union of complete balanced bipartite graphs.