# Equivelar and d-Covered Triangulations of Surfaces. I - Mathematics > Combinatorics

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Abstract: We survey basic properties and bounds for $q$-equivelar and $d$-coveredtriangulations of closed surfaces. Included in the survey is a list of theknown sources for $q$-equivelar and $d$-covered triangulations. We identify allorientable and non-orientable surfaces $M$ of Euler characteristic$0>\chiM\geq -230$ which admit non-neighborly $q$-equivelar triangulationswith equality in the upper bound$q\leq\Bigl\lfloor\tfrac{1}{2}5+\sqrt{49-24\chi M}\Bigl floor$. Theseexamples give rise to $d$-covered triangulations with equality in the upperbound $d\leq2\Bigl\lfloor\tfrac{1}{2}5+\sqrt{49-24\chi M}\Bigl floor$. Ageneralization of Ringel-s cyclic $7{ m mod}12$ series of neighborlyorientable triangulations to a two-parameter family of cyclic orientabletriangulations $R {k,n}$, $k\geq 0$, $n\geq 7+12k$, is the main result of thispaper. 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 forthe upper bound for $q$ as well as derived examples with equality for the upperbound for $d$.

Author: ** Frank H. Lutz, Thom Sulanke, Anand K. Tiwari, Ashish K. Upadhyay**

Source: https://arxiv.org/