摘要：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/nâ^^ (0,1/2). Given g,nâ^^ â". with gâ' â^Z and n/2-gâ' â^Z as nâ' â^Z, the number C_{n,g} of cubic maps on Σ_g with 2n vertices satisfies C_{n,g} â^¼ (g!)² α(θ) β(θ)⿠γ(θ)^{2g}, as gâ' â^Z, 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/nâ' 0 or g/nâ' 1/2.
关键词：cubic maps; triangulations; cubic graphs on surfaces; generating functions; asymptotic enumeration; local limit theorem; saddle-point method
