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 $0>\chi(M)\geq -230$ which admit non-neighborly $q$-equivelar triangulations with equality in the upper bound $q\leq\Bigl\lfloor\tfrac{1}{2}(5+\sqrt{49-24\chi (M)})\Bigl\rfloor$. These examples give rise to $d$-covered triangulations with equality in the upper bound $d\leq2\Bigl\lfloor\tfrac{1}{2}(5+\sqrt{49-24\chi (M)})\Bigl\rfloor$. A generalization of Ringel's cyclic $7{\rm mod}12$ series of neighborly orientable triangulations to a two-parameter family of cyclic orientable triangulations $R_{k,n}$, $k\geq 0$, $n\geq 7+12k$, is the main result of this paper. In particular, the two infinite subseries $R_{k,7+12k+1}$ and $R_{k,7+12k+2}$, $k\geq 1$, provide non-neighborly examples with equality for the upper bound for $q$ as well as derived examples with equality for the upper bound for $d$.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2010
- DOI:
- 10.48550/arXiv.1001.2777
- arXiv:
- arXiv:1001.2777
- Bibcode:
- 2010arXiv1001.2777L
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Geometric Topology;
- 05C10;
- 57Q15;
- 52B70;
- 52B05;
- 57M20
- E-Print:
- 21 pages, 4 figures