Short survey on stable polynomials, orientations and matchings

TL;DR

This survey explores the theory of stable polynomials and their applications, providing proofs of key theorems related to matchings and orientations in regular graphs, and offering a unified generalization.

Contribution

It offers self-contained proofs of two important theorems using stable polynomial theory and presents a unified generalization of these results.

Findings

Proof of lower bounds on perfect matchings in regular bipartite graphs.

Proof of lower bounds on Eulerian orientations in regular graphs.

Unified framework connecting matchings and orientations via stable polynomials.

Abstract

This is a short survey about the theory of stable polynomials and its applications. It gives self-contained proofs of two theorems of Schrijver. One of them asserts that for a $d$--regular bipartite graph $G$ on $2n$ vertices, the number of perfect matchings, denoted by $\mathrm{pm}(G)$, satisfies $$\mathrm{pm}(G)\geq \bigg( \frac{(d-1)^{d-1}}{d^{d-2}} \bigg)^{n}.$$ The other theorem claims that for even $d$ the number of Eulerian orientations of a $d$--regular graph $G$ on $n$ vertices, denoted by $\varepsilon(G)$, satisfies $$\varepsilon(G)\geq \bigg(\frac{\binom{d}{d/2}}{2^{d/2}}\bigg)^n.$$ To prove these theorems we use the theory of stable polynomials, and give a common generalization of the two theorems.