Counting Cubic Maps with Large Genus

Z. Gao, M. Kang

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We derive an asymptotic expression for the number of cubic maps on orientable surfaces when the genus is proportional to the number of vertices. Let Σ_g denote the orientable surface of genus g and θ=g/n∈ (0,1/2). Given g,n∈ ℕ with g→ ∞ and n/2-g→ ∞ as n→ ∞, the number C_{n,g} of cubic maps on Σ_g with 2n vertices satisfies C_{n,g} ∼ (g!)² α(θ) β(θ)ⁿ γ(θ)^{2g}, as g→ ∞, where α(θ),β(θ),γ(θ) are differentiable functions in (0,1/2). This also leads to the asymptotic number of triangulations (as the dual of cubic maps) with large genus. When g/n lies in a closed subinterval of (0,1/2), the asymptotic formula can be obtained using a local limit theorem. The saddle-point method is applied when g/n→ 0 or g/n→ 1/2.
Original languageEnglish
Title of host publication31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms
EditorsMichael Drmota, Clemens Heuberger
Place of PublicationSaarbrücken/Wadern
PublisherSchloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH
Number of pages13
ISBN (Electronic)9783959771474
ISBN (Print)18688969
DOIs
Publication statusPublished - 2020
Event31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms - https://www.math.aau.at/AofA2020/, Virtuell, Austria
Duration: 15 Jun 202019 Jun 2020

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume159

Conference

Conference31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms
Abbreviated titleAofA2020
CountryAustria
CityVirtuell
Period15/06/2019/06/20

Keywords

  • Asymptotic enumeration
  • Cubic graphs on surfaces
  • Cubic maps
  • Generating functions
  • Local limit theorem
  • Saddle-point method
  • Triangulations

ASJC Scopus subject areas

  • Software

Fields of Expertise

  • Information, Communication & Computing

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