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On the Maximum Degree of a Random Planar Graph

Published online by Cambridge University Press:  01 July 2008

COLIN McDIARMID  and
BRUCE REED
COLIN McDIARMID
Affiliation:
Department of Statistics, University of Oxford, UK (e-mail: cmcd@stats.ox.ac.uk)
BRUCE REED
Affiliation:
Canada Research Chair in Graph Theory, Department of Computer Science, McGill University, Montreal, Canada and Laboratoire I3S, CNRS Sophia-Antipolis, France (e-mail: breed@cs.mcgill.ca)
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Abstract

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Let the random graph Rn be drawn uniformly at random from the set of all simple planar graphs on n labelled vertices. We see that with high probability the maximum degree of Rn is Θ(ln n). We consider also the maximum size of a face and the maximum increase in the number of components on deleting a vertex. These results extend to graphs embeddable on any fixed surface.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 17 , Issue 4 , July 2008 , pp. 591 - 601
Copyright
Copyright © Cambridge University Press 2008

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