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A short survey on stable polynomials, orientations and matchings

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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.

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Acknowledgements

The first author thanks Jonathan Leake for the discussions on the topic of this paper. The authors are very grateful to the anonymous referee for his/her suggestions that greatly improved the paper.

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Authors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, H-1053, Budapest, Hungary

    P. Csikvári

  2. Mathematics Institute, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary

    P. Csikvári & Á. Schweitzer

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  1. P. Csikvári
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  2. Á. Schweitzer
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Correspondence to P. Csikvári.

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The first author is supported by the Counting in Sparse Graphs Lendület Research Group of the Alfréd Rényi Institute of Mathematics.

The second author is partially supported by the EFOP program (EFOP-3.6.3-VEKOP-16-2017-00002).

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Csikvári, P., Schweitzer, Á. A short survey on stable polynomials, orientations and matchings. Acta Math. Hungar. 166, 1–16 (2022). https://doi.org/10.1007/s10474-022-01208-3

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  • DOI: https://doi.org/10.1007/s10474-022-01208-3

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