TY - GEN

T1 - Counting Cubic Maps with Large Genus

AU - Gao, Z.

AU - Kang, M.

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - Asymptotic enumeration

KW - Cubic graphs on surfaces

KW - Cubic maps

KW - Generating functions

KW - Local limit theorem

KW - Saddle-point method

KW - Triangulations

UR - http://www.scopus.com/inward/record.url?eid=2-s2.0-85091029693&partnerID=MN8TOARS

U2 - 10.4230/LIPIcs.AofA.2020.13

DO - 10.4230/LIPIcs.AofA.2020.13

M3 - Conference contribution

SN - 18688969

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms

A2 - Drmota, Michael

A2 - Heuberger, Clemens

PB - Schloss Dagstuhl, Leibniz-Zentrum fü Informatik GmbH

CY - Saarbrücken/Wadern

ER -