Equivelar and d-Covered Triangulations of Surfaces. I

الأصل · EN

We survey basic properties and bounds for q-equivelar and d-covered triangulations of closed surfaces. Included in the survey is a list of the known sources for q-equivelar and d-covered triangulations. We identify all orientable and non-orientable surfaces M of Euler characteristic 0>χ(M)≥ -230 which admit non-neighborly q-equivelar triangulations with equality in the upper bound q≤12(5+√49-24χ(M)). These examples give rise to d-covered triangulations with equality in the upper bound d≤212(5+√49-24χ(M)). A generalization of Ringel's cyclic 7 mod12 series of neighborly orientable triangulations to a two-parameter family of cyclic orientable triangulations Rₖ,ₙ, k≥ 0, n≥ 7+12k, is the main result of this paper. In particular, the two infinite subseries Rₖ,₇₊₁₂ₖ₊₁ and Rₖ,₇₊₁₂ₖ₊₂, k≥ 1, provide non-neighborly examples with equality for the upper bound for q as well as derived examples with equality for the upper bound for d.