[Paper Review] Equivelar and d-Covered Triangulations of Surfaces. II. Cyclic Triangulations and Tessellations

Frank H. Lutz|arXiv (Cornell University)|Jan 15, 2010

Advanced Combinatorial Mathematics12 references4 citations

TL;DR

This paper introduces an infinite family of cyclic triangulations—specifically the [0,1,2]-family—that yield vertex-transitive, q-equi-regular triangulations of both orientable and non-orientable surfaces for all q = 3k (k ≥ 2) and q = 3k+1 (k ≥ 3). It constructs explicit two-parameter series of such triangulations and derives corresponding cyclic tessellations, proving the existence of infinite series of q-equi-regular triangulations beyond previously known examples, including the first systematic construction for q = 7.

ABSTRACT

With the $[0,1,2]$-family of cyclic triangulations we introduce a rich class of vertex-transitive triangulations of surfaces. In particular, there are infinite series of cyclic $q$-equivelar triangulations of orientable and non-orientable surfaces for every $q=3k$, $k\geq 2$, and every $q=3k+1$, $k\geq 3$. Series of cyclic tessellations of surfaces are derived from these triangulated series.

Motivation & Objective

• To construct explicit infinite families of vertex-transitive, q-equi-regular triangulations of surfaces for previously unexplored values of q.
• To extend known cyclic triangulation series beyond the classical Ringel and Altshuler constructions.
• To provide a systematic method for generating cyclic triangulations and their derived tessellations using the [0,1,2]-family framework.
• To resolve the existence question for q-equi-regular triangulations by constructing infinite series for all admissible q ≥ 6 except q = 5 and q = 6.
• To describe fundamental domains and derive tessellations from triangulations, including strongly regular cell decompositions via edge deletion from seed patterns.

Proposed method

• The [0,1,2]-family of cyclic triangulations is defined via specific difference sets in cyclic groups, generating vertex-transitive triangulations with controlled vertex degrees.
• Two-parameter series S_{k,n}, T_{k,n}, ..., Z_{k,n} are constructed using cyclic symmetry and difference sets to ensure q-equi-regularity for q = 3k or q = 3k+1.
• The f-vector of a q-equi-regular triangulation is derived as f = (n, nq/2, nq/3), linking the number of vertices n to the vertex degree q and Euler characteristic.
• Fundamental domains are analyzed to derive tessellations, with edge deletion from seed patterns yielding (s+2)-gonal cell decompositions.
• For q = 7, the paper proves there are exactly two cyclic 7-equi-regular triangulations on the genus-2 surface with 12 vertices.
• Generalizations of known polyhedral maps (e.g., Brehm’s {5,5}-maps) are extended to infinite series D_{ {2k+1,2k+1} }(n) and D_{ {2k,2k} }(n) via difference set constructions.

Experimental results

Research questions

• RQ1 Can infinite families of cyclic q-equi-regular triangulations be constructed for all q = 3k (k ≥ 2) and q = 3k+1 (k ≥ 3)?
• RQ2 What is the structure and existence condition for cyclic triangulations of surfaces with vertex-transitive symmetry beyond known series?
• RQ3 How can cyclic triangulations be systematically transformed into tessellations with strongly regular cell decompositions?
• RQ4 What are the necessary and sufficient conditions for a cyclic triangulation to yield a polyhedral map with regular face and vertex patterns?
• RQ5 For which values of q and n do cyclic q-equi-regular triangulations of surfaces exist, and can they be explicitly parameterized?

Key findings

• The paper constructs infinite series of cyclic q-equi-regular triangulations for all q = 3k with k ≥ 2 and q = 3k+1 with k ≥ 3, proving their existence on both orientable and non-orientable surfaces.
• For q = 7, there are exactly two cyclic 7-equi-regular triangulations, both on the orientable genus-2 surface with 12 vertices.
• The two-parameter series S_{k,n}, T_{k,n}, ..., Z_{k,n} are explicitly defined and shown to yield q-equi-regular triangulations for q = 3k or q = 3k+1.
• The [0,1,2]-family provides a unifying framework for generating vertex-transitive triangulations, with all such triangulations falling into q = 3k or q = 3k+1 types.
• By deleting orbits of interior edges from seed patterns, the paper derives strongly regular tessellations, including generalizations of Brehm’s {5,5}-maps and {6,6}-maps.
• The series D_{ {2k+1,2k+1} }(n) and D_{ {2k,2k} }(n) are shown to be orientable for even r and non-orientable for odd r, extending known polyhedral map constructions.