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2022 On the expected number of perfect matchings in cubic planar graphs
Marc Noy, Clément Requilé, Juanjo Rué
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Publ. Mat. 66(1): 325-353 (2022). DOI: 10.5565/PUBLMAT6612213


A well-known conjecture by Lovász and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. ([13]). On the other hand, Chudnovsky and Seymour ([8]) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with n vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically cγn, where c>0 and γ~1.14196 is an explicit algebraic number. We also compute the expected number of perfect matchings in (not necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.


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Marc Noy. Clément Requilé. Juanjo Rué. "On the expected number of perfect matchings in cubic planar graphs." Publ. Mat. 66 (1) 325 - 353, 2022.


Received: 27 May 2020; Accepted: 25 February 2021; Published: 2022
First available in Project Euclid: 4 January 2022

Digital Object Identifier: 10.5565/PUBLMAT6612213

Primary: 05A16

Keywords: Asymptotic enumeration , Planar graphs , regular graphs

Rights: Copyright © 2022 Universitat Autònoma de Barcelona, Departament de Matemàtiques


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Vol.66 • No. 1 • 2022
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