Equivelar and d-Covered Triangulations of Surfaces. I

Abstract

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 $0>\chi (M)\ge -230$ which admit non-neighborly $q$-equivelar triangulations with equality in the upper bound $q\le \lfloor \frac{1}{2}(5+\sqrt{49-24\chi (M)})\rfloor$. These examples give rise to $d$-covered triangulations with equality in the upper bound $d\le 2\lfloor \frac{1}{2}(5+\sqrt{49-24\chi (M)})\rfloor$. A generalization of Ringel's cyclic $7\mathrm{mod}12$ series of neighborly orientable triangulations to a two-parameter family of cyclic orientable triangulations $R_{k,n}$, $k\ge 0$, $n\ge 7+12k$, is the main result of this paper. In particular, the two infinite subseries $R_{k,7+12k+1}$ and $R_{k,7+12k+2}$, $k\ge 1$, provide non-neighborly examples with equality for the upper bound for $q$ as well as derived examples with equality for the upper bound for $d$.

21 pages, 4 figures

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Cited by in corpus (3)

• Equivelar and d-Covered Triangulations of Surfaces. II. Cyclic Triangulations and Tessellations
• Stanley-Reisner rings for symmetric simplicial complexes, G-semimatroids and Abelian arrangements
• Semi-equivelar maps on the surface of Euler characteristic -1