dc.contributor.author Noy Serrano, Marcos dc.contributor.author Ravelomanana, Vlady dc.contributor.author Rue, Juanjo dc.contributor.other Universitat Politècnica de Catalunya. Departament de Matemàtiques dc.date.accessioned 2016-03-02T18:49:21Z dc.date.available 2016-03-02T18:49:21Z dc.date.issued 2015-03-01 dc.identifier.citation Noy, 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.issn 0002-9939 dc.identifier.uri http://hdl.handle.net/2117/83733 dc.description.abstract Let 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.extent 12 p. dc.language.iso eng dc.rights.uri http://creativecommons.org/licenses/by-nc-nd/3.0/es/ dc.subject Àrees temàtiques de la UPC::Matemàtiques i estadística dc.subject.other asymptotic enumeration dc.subject.other transition dc.subject.other evolution dc.title On the probability of planarity of a random graph near the critical point dc.type Article dc.contributor.group Universitat Politècnica de Catalunya. MD - Matemàtica Discreta dc.description.peerreviewed Peer Reviewed dc.relation.publisherversion http://www.ams.org/journals/proc/2015-143-03/S0002-9939-2014-12141-1/S0002-9939-2014-12141-1.pdf dc.rights.access Open Access drac.iddocument 15597913 dc.description.version Postprint (author's final draft) upcommons.citation.author Noy, M., Ravelomanana, V., Rue, J. upcommons.citation.published true upcommons.citation.publicationName Proceedings of the American Mathematical Society upcommons.citation.volume 143 upcommons.citation.number 3 upcommons.citation.startingPage 925 upcommons.citation.endingPage 936
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