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 $\mathrm{pm}(G)$, 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.