A short survey on stable polynomials, orientations and matchings

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 \({\rm pm}(G)\), satisfies $${\rm pm}(G)\geq \Bigl( \frac{(d-1)^{d-1}}{d^{d-2}} \Bigr)^{n}.$$ The other theorem claims that for even \(d\) the number of Eulerian orientations of a \(d\)-regular graph \(G\) on \(n\) vertices, denoted by \(\epsilon(G)\), satisfies $$\epsilon(G)\geq \biggl(\frac{\binom{d}{d/2}}{2^{d/2}}\biggr)^n.$$ To prove these theorems we use the theory of stable polynomials, and give a common generalization of the two theorems.