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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.AofA.2020.13
URN: urn:nbn:de:0030-drops-120437
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12043/
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Gao, Zhicheng ; Kang, Mihyun

Counting Cubic Maps with Large Genus

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LIPIcs-AofA-2020-13.pdf (0.9 MB)

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.

BibTeX - Entry

@InProceedings{gao_et_al:LIPIcs:2020:12043,
  author =	{Zhicheng Gao and Mihyun Kang},
  title =	{{Counting Cubic Maps with Large Genus}},
  booktitle =	{31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)},
  pages =	{13:1--13:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-147-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{159},
  editor =	{Michael Drmota and Clemens Heuberger},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12043},
  URN =		{urn:nbn:de:0030-drops-120437},
  doi =		{10.4230/LIPIcs.AofA.2020.13},
  annote =	{Keywords: cubic maps, triangulations, cubic graphs on surfaces, generating functions, asymptotic enumeration, local limit theorem, saddle-point method}
}

Keywords: cubic maps, triangulations, cubic graphs on surfaces, generating functions, asymptotic enumeration, local limit theorem, saddle-point method
Collection: 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020)
Issue Date: 2020
Date of publication: 10.06.2020

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