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Limit laws of planar maps with prescribed vertex degrees

Part of: Combinatorics Graph theory

Published online by Cambridge University Press:  04 February 2019

G. Collet ,
M. Drmota  and
L. D. Klausner
G. Collet
Affiliation:
TU Wien, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8–10, A-1040 Wien, Austria
M. Drmota*
Affiliation:
TU Wien, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8–10, A-1040 Wien, Austria
L. D. Klausner
Affiliation:
TU Wien, Institute of Discrete Mathematics and Geometry, Wiedner Hauptstraße 8–10, A-1040 Wien, Austria
*
*Corresponding author. Email: michael.drmota@tuwien.ac.at
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Abstract

We prove a generalmulti-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. Our results rely on a classical bijection with mobiles (objects exhibiting a tree structure), combined with refined analytic tools to deal with the systems of equations on infinite variables that arise. We also discuss possible extensions to maps of higher genus and to weighted maps.

MSC classification

Primary: 05A16: Asymptotic enumeration
Secondary: 05C07: Vertex degrees 05C10: Planar graphs; geometric and topological aspects of graph theory 05C30: Enumeration in graph theory
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 28 , Special Issue 4: Analysis of Algorithms , July 2019 , pp. 519 - 541
Copyright
© Cambridge University Press 2019 

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Footnotes

†

Partially supported by the Austrian Science Fund (FWF) project SFB F50-02 ‘Shape Characteristics of Planar Maps and Planar Graphs’.

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