[Paper Review] Equivelar and d-Covered Triangulations of Surfaces. I

Frank H. Lutz, T. Sulanke+2 more|arXiv (Cornell University)|Jan 15, 2010

Advanced Combinatorial Mathematics41 references3 citations

TL;DR

This paper introduces a two-parameter family of cyclic, orientable triangulations $ R_{k,n} $, generalizing Ringel's neighborly series, which achieve equality in the upper bound for $ q $-equivelar triangulations. It proves that the subseries $ R_{k,7+12k+1} $ and $ R_{k,7+12k+2} $ yield non-neighborly examples with maximal $ q = 6+12k $, and their subdivisions produce $ 2q $-covered triangulations achieving equality in the $ d $-covered bound for surfaces with $ \chi(M) \geq -230 $.

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 $0>χ(M)\geq -230$ which admit non-neighborly $q$-equivelar triangulations with equality in the upper bound $q\leq\Bigl\lfloor frac{1}{2}(5+\sqrt{49-24χ(M)})\Bigl floor$. These examples give rise to $d$-covered triangulations with equality in the upper bound $d\leq2\Bigl\lfloor frac{1}{2}(5+\sqrt{49-24χ(M)})\Bigl floor$. A generalization of Ringel's cyclic $7{ m mod}12$ 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 $d$.

Motivation & Objective

• To extend Ringel’s cyclic neighborly triangulations to a broader family of non-neighborly, equivelar triangulations.
• To identify all orientable and non-orientable surfaces with $ \chi(M) \geq -230 $ that admit $ q $-equivelar triangulations achieving the theoretical upper bound on $ q $.
• To construct $ d $-covered triangulations from these $ q $-equivelar examples, achieving equality in the $ d $-covered bound.
• To provide a systematic survey of known sources and bounds for $ q $-equivelar and $ d $-covered triangulations.
• To establish vertex-minimal, vertex-transitive triangulations for orientable surfaces via the $ R_{k,n} $ series.

Proposed method

• Generalize Ringel’s cyclic neighborly triangulations $ R_{k,7+12k} $ to a two-parameter family $ R_{k,n} $ for $ n \geq 7+12k $, $ k \geq 0 $, using cyclic group action on generating triangles.
• Define $ R_{k,n} $ via a fixed set of $ 2+4k $ generating triangles under $ \mathbb{Z}_n $-action, preserving the link structure of vertex 0 for $ n = 7+12k $.
• Prove orientability by showing compatibility of vertex-star orientations via Ringel’s rule $ R^* $, which holds under cyclic shifts for $ n \geq 7+12k $.
• Use the identity $ \chi(M) = -2kn $ for $ R_{k,n} $ to compute Euler characteristic and verify $ q $-equivelar type $ \{3,6+12k\} $ with $ f $-vector $ (n, 3(1+2k)n, 2(1+2k)n) $.
• Apply vertex subdivision to $ q $-equivelar triangulations to produce $ 2q $-covered triangulations, inheriting equality in the $ d $-covered bound.
• Verify that $ R_{k,7+12k+1} $ and $ R_{k,7+12k+2} $ achieve $ q = \lfloor \frac{1}{2}(5 + \sqrt{49 - 24\chi}) \rfloor $, confirming equality in the upper bound.

Experimental results

Research questions

• RQ1 Which orientable and non-orientable surfaces with $ \chi(M) \geq -230 $ admit $ q $-equivelar triangulations achieving the theoretical upper bound $ q \leq \lfloor \frac{1}{2}(5 + \sqrt{49 - 24\chi}) \rfloor $?
• RQ2 Can the Ringel cyclic neighborly series be generalized to non-neighborly examples that still achieve equality in the $ q $-equivelar bound?
• RQ3 Do the resulting $ q $-equivelar triangulations yield $ d $-covered triangulations with equality in the $ d $-covered upper bound $ d \leq 2\lfloor \frac{1}{2}(5 + \sqrt{49 - 24\chi}) \rfloor $?
• RQ4 Are there vertex-minimal, vertex-transitive triangulations of orientable surfaces with $ \chi(M) = -2kn $, and do they arise from the generalized $ R_{k,n} $ family?
• RQ5 What conditions ensure orientability of the generalized $ R_{k,n} $ triangulations for $ n \geq 7+12k $?

Key findings

• The generalized Ringel series $ R_{k,n} $, for $ k \geq 0 $, $ n \geq 7+12k $, forms a two-parameter family of cyclic, orientable triangulations of genus $ g = kn + 1 $ with $ f $-vector $ (n, 3(1+2k)n, 2(1+2k)n) $.
• The subseries $ R_{k,7+12k+1} $ and $ R_{k,7+12k+2} $, for $ k \geq 1 $, achieve equality in the upper bound $ q \leq \lfloor \frac{1}{2}(5 + \sqrt{49 - 24\chi}) \rfloor $, providing non-neighborly $ q $-equivelar triangulations.
• These $ R_{k,7+12k+1} $ and $ R_{k,7+12k+2} $ examples yield $ 2q $-covered triangulations that achieve equality in the upper bound $ d \leq 2\lfloor \frac{1}{2}(5 + \sqrt{49 - 24\chi}) \rfloor $.
• For $ k \geq 1 $, the triangulations $ R_{k,7+12k+1} $ are vertex-minimal for orientable surfaces with $ \chi = -2kn $, as they meet the Heawood bound with equality.
• The orientability of $ R_{k,n} $ for $ n \geq 7+12k $ is established via compatibility of vertex-star orientations under cyclic symmetry, verified using Ringel’s rule $ R^* $.
• The construction confirms Conjecture 8 for $ q = 6+12k $, $ k \geq 1 $, by providing explicit examples of $ q $-equivelar triangulations achieving the theoretical maximum $ q $ for surfaces with $ \chi = -2kn $.