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On the Diameter of Random Planar Graphs

Published online by Cambridge University Press:  18 September 2014

GUILLAUME CHAPUY ,
ÉRIC FUSY ,
OMER GIMÉNEZ  and
MARC NOY
GUILLAUME CHAPUY
Affiliation:
CNRS, LIAFA, UMR 7089, Université Paris Diderot–Paris 7, Case 7014, 75205 Paris CEDEX 13, France (e-mail: guillaume.chapuy@liafa.univ-paris-diderot.fr)
ÉRIC FUSY
Affiliation:
CNRS, LIX, UMR 7161, École Polytechnique, 91128 Palaiseau CEDEX, France (e-mail: fusy@lix.polytechnique.fr)
OMER GIMÉNEZ
Affiliation:
Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Barcelona, Spain (e-mail: omer.gimenez@gmail.com)
MARC NOY
Affiliation:
Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain (e-mail: marc.noy@upc.edu)
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Abstract

We show that the diameter diam(Gn) of a random labelled connected planar graph with n vertices is equal to n1/4+o(1), in probability. More precisely, there exists a constant c > 0 such that

$$ P(\D(G_n)\in(n^{1/4-\e},n^{1/4+\e}))\geq 1-\exp(-n^{c\e}) $$
for ε small enough and n ≥ n0(ε). We prove similar statements for 2-connected and 3-connected planar graphs and maps.

Keywords

Primary 05C80Secondary 05C3005A1682B4168Q87
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015 , pp. 145 - 178
Copyright
Copyright © Cambridge University Press 2014 

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