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Combinatorics, Complexity, and ChanceA Tribute to Dominic Welsh$
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Geoffrey Grimmett and Colin McDiarmid

Print publication date: 2007

Print ISBN-13: 9780198571278

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198571278.001.0001

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RANDOM PLANAR GRAPHS WITH GIVEN AVERAGE DEGREE

RANDOM PLANAR GRAPHS WITH GIVEN AVERAGE DEGREE

Chapter:
(p.83) 6 RANDOM PLANAR GRAPHS WITH GIVEN AVERAGE DEGREE
Source:
Combinatorics, Complexity, and Chance
Author(s):

Stefanie Gerke

Colin McDiarmid

Angelika Steger

Andreas Weißl

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198571278.003.0006

This chapter investigates the behaviour of the function γ(q). In particular, it considers γ(q) as q approaches 1 from below and above and 3 from below, and explains the discontinuity as it approach 1 from below by changing scale appropriately. New properties of Rn,q, for example in terms of the number of automorphisms, are investigated. Lower bounds are given on the maximum node degree and the maximum face size of a plane embedding. The relation between the growth constants of labelled and unlabelled planar graphs with given average degree is studied.

Keywords:   planar graphs, unlabelled graphs, automorphisms, growth constants

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