Random cubic planar graphs

Authors:
Manuel Bodirsky;Mihyun Kang;Mike Löffler;Colin McDiarmid
Affiliations:
Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, D-10099 Berlin, Germany;Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, D-10099 Berlin, Germany;Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, D-10099 Berlin, Germany;University of Oxford, Department of Statistics, 1 South Parks Road, Oxford OX 1 3TG, United Kingdom
Venue:
Random Structures & Algorithms - Proceedings from the 12th International Conference “Random Structures and Algorithms”, August1-5, 2005, Poznan, Poland
Year:
2007

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Abstract

We show that the number of labeled cubic planar graphs on nvertices with n even is asymptoticallyαn-7/2ρ-nn!,where ρ-1 ... 3.13259 and α are analyticconstants. We show also that the chromatic number of a random cubicplanar graph that is chosen uniformly at random among all thelabeled cubic planar graphs on n vertices is three withprobability tending to e-ρ4/4! ... 0.999568and four with probability tending to 1 -e-ρ4/4! as n → ∞ withn even. The proof given combines generating functiontechniques with probabilistic arguments. © 2006 WileyPeriodicals, Inc. Random Struct. Alg., 2007