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dc.contributor.authorNoy Serrano, Marcos
dc.contributor.authorRavelomanana, Vlady
dc.contributor.authorRue, Juanjo
dc.contributor.otherUniversitat Politècnica de Catalunya. Departament de Matemàtiques
dc.date.accessioned2016-03-02T18:49:21Z
dc.date.available2016-03-02T18:49:21Z
dc.date.issued2015-03-01
dc.identifier.citationNoy, M., Ravelomanana, V., Rue, J. On the probability of planarity of a random graph near the critical point. "Proceedings of the American Mathematical Society", 01 Març 2015, vol. 143, núm. 3, p. 925-936.
dc.identifier.issn0002-9939
dc.identifier.urihttp://hdl.handle.net/2117/83733
dc.description.abstractLet G(n, M) be the uniform random graph with n vertices and M edges. Erdos and Renyi (1960) conjectured that the limiting probability; lim(n ->infinity) Pr{G(n, n/2) is planar}; exists and is a constant strictly between 0 and 1. Luczak, Pittel and Wierman (1994) proved this conjecture, and Janson, Luczak, Knuth and Pittel (1993) gave lower and upper bounds for this probability. In this paper we determine the exact limiting probability of a random graph being planar near the critical point M = n/2. For each lambda, we find an exact analytic expression for; p(lambda) = lim(n ->infinity) Pr {G (n, n/2 (1 + lambda(n-1/3))) is planar}.; In particular, we obtain p(0) approximate to 0.99780. We extend these results to classes of graphs closed under taking minors. As an example, we show that the probability of G(n, n/2) being series-parallel converges to 0.98003. For the sake of completeness and exposition we reprove in a concise way several basic properties we need of a random graph near the critical point.
dc.format.extent12 p.
dc.language.isoeng
dc.rights.urihttp://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.subjectÀrees temàtiques de la UPC::Matemàtiques i estadística
dc.subject.otherasymptotic enumeration
dc.subject.othertransition
dc.subject.otherevolution
dc.titleOn the probability of planarity of a random graph near the critical point
dc.typeArticle
dc.contributor.groupUniversitat Politècnica de Catalunya. MD - Matemàtica Discreta
dc.description.peerreviewedPeer Reviewed
dc.relation.publisherversionhttp://www.ams.org/journals/proc/2015-143-03/S0002-9939-2014-12141-1/S0002-9939-2014-12141-1.pdf
dc.rights.accessOpen Access
drac.iddocument15597913
dc.description.versionPostprint (author's final draft)
upcommons.citation.authorNoy, M., Ravelomanana, V., Rue, J.
upcommons.citation.publishedtrue
upcommons.citation.publicationNameProceedings of the American Mathematical Society
upcommons.citation.volume143
upcommons.citation.number3
upcommons.citation.startingPage925
upcommons.citation.endingPage936


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