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Counting Cubic Maps with Large Genus

Authors Zhicheng Gao, Mihyun Kang



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Author Details

Zhicheng Gao
  • School of Mathematics and Statistics, Carleton University, Ottawa, Canada
Mihyun Kang
  • Institute of Discrete Mathematics, Graz University of Technology, Austria

Acknowledgements

Part of this research was carried out while the first author was visiting TU Graz. This visit was financially supported by TU Graz within the Doctoral Program "Discrete Mathematics".

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Zhicheng Gao and Mihyun Kang. Counting Cubic Maps with Large Genus. In 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 159, pp. 13:1-13:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.AofA.2020.13

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
Keywords
  • cubic maps
  • triangulations
  • cubic graphs on surfaces
  • generating functions
  • asymptotic enumeration
  • local limit theorem
  • saddle-point method

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References

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