#
Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K_{2,2,2,2}

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary: The Orbit Decomposition

- (i)
- the set of n-vertex trees (Section 4) and
- (ii)

## 3. A New Enumerative Polynomial

**Theorem**

**1.**

- (I)
- $P\left(1\right)=\left|\mathsf{\Omega}\right|$,
- (II)
- ${P}^{\prime}\left(1\right)=\tilde{\alpha}$, the total number of nontrivial automorphisms when counted across all elements of Ω,
- (III)
- $\underset{0}{\overset{1}{\int}}P\left(x\right)dx=\left|\mathsf{\Lambda}\right|/\left|\mathsf{\Gamma}\right|$.

**Proof.**

- (I)
- Evaluate at $x=1$:$$P\left(1\right)=\sum _{k}|{\mathsf{\Omega}}_{k}|=|\mathsf{\Omega}|.$$

- (II)
- Evaluate the derivative:$${P}^{\prime}\left(x\right)=\sum _{k\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}\right|\mathsf{\Gamma}|}(k-1)\left|{\mathsf{\Omega}}_{k}\right|\phantom{\rule{0.166667em}{0ex}}{x}^{k-2},$$$${P}^{\prime}\left(1\right)=\sum _{k\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}\right|\mathsf{\Gamma}|}(k-1)|{\mathsf{\Omega}}_{k}|=\sum _{k\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}\right|\mathsf{\Gamma}|}k|{\mathsf{\Omega}}_{k}|-|\mathsf{\Omega}|.$$Therefore,$${P}^{\prime}\left(1\right)=\tilde{\alpha}=\alpha -\overline{\alpha},$$

- (III)
- Evaluate the integral:$$\underset{0}{\overset{1}{\int}}P\left(x\right)dx=\sum _{k\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}\right|\mathsf{\Gamma}|}\frac{1}{k}\left|{\mathsf{\Omega}}_{k}\right|.$$Thus, by Equation (2):$$\left|\mathsf{\Gamma}\right|\underset{0}{\overset{1}{\int}}P\left(x\right)dx=\sum _{k\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}\right|\mathsf{\Gamma}|}\frac{\left|\mathsf{\Gamma}\right|}{k}|{\mathsf{\Omega}}_{k}|=\sum _{k\phantom{\rule{0.166667em}{0ex}}\left|\phantom{\rule{0.166667em}{0ex}}\right|\mathsf{\Gamma}|}|{\mathsf{\Lambda}}_{k}|=|\mathsf{\Lambda}|.$$Thus,$$\underset{0}{\overset{1}{\int}}P\left(x\right)dx=\frac{\left|\mathsf{\Lambda}\right|}{\left|\mathsf{\Gamma}\right|}.$$

## 4. Examples: Trees

#### 4.1. Trees with 4 Vertices

#### 4.2. Trees with 5 Vertices

## 5. Symmetry Properties of the Graph $\mathit{G}={\mathit{K}}_{\mathbf{2},\mathbf{2},\mathbf{2},\mathbf{2}}$ and Its Triangular Embeddings in the Torus

## 6. Systematic Generation of Triangulations of the Torus with the Vertex-Labeled Graph $\mathit{G}={\mathit{K}}_{\mathbf{2},\mathbf{2},\mathbf{2},\mathbf{2}}$

**Lemma**

**1.**

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 7. Conclusive Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Lawrencenko, S.; Vyalyi, M.N.; Zgonnik, L.V. Grünbaum coloring and its generalization to arbitrary dimension. Australas. J. Combin.
**2017**, 67, 119–130. [Google Scholar] - Maslova, Y.V.; Petrov, M.V. Lavrenchenko’s polyhedron of genus one. In Some Actual Problems of Modern Mathematics and Mathematical Education; Herzen Readings—2018 St. Petersburg (9–13 April 2018); Russian Herzen State Pedagogical University: St. Petersburg, Russia, 2018; pp. 162–168. (In Russian) [Google Scholar]
- Schaller, D.; Geiss, M.; Hellmuth, M.; Stadler, P.F. Arc-completion of 2-colored best match graphs to binary-explainable best match graphs. Algorithms
**2021**, 14, 110. [Google Scholar] [CrossRef] - Stanković, L.; Lerga, J.; Mandic, D.; Brajović, M.; Richard, C.; Daković, M. From time-frequency to vertex-frequency and back. Mathematics
**2021**, 9, 1407. [Google Scholar] [CrossRef] - Tomescu, M.A.; Jäntschi, L.; Rotaru, D.I. Figures of graph partitioning by counting, sequence and layer matrices. Mathematics
**2021**, 9, 1419. [Google Scholar] [CrossRef] - Cayley, A. A theorem on trees. Quart. J. Pure Appl. Math.
**1889**, 23, 376–378. [Google Scholar] - Aigner, M.; Ziegler, G.M. Cayley’s formula for the number of trees. In Proofs from The Book; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Otter, R. The number of trees. Ann. Math.
**1948**, 49, 583–599. [Google Scholar] [CrossRef] - Sloane, N.J.A. OEIS Sequence A000055: Number of Trees with n Unlabeled Nodes. The On-Line Encyclopedia of Integer Sequences. Available online: https://oeis.org/A000055 (accessed on 2 August 2021).
- Lang, S. Algebra, 3rd ed.; Springer: New York, NY, USA, 2002. [Google Scholar]
- Lawrencenko, S. Irreducible triangulations of the torus. Ukr. Geom. Sb.
**1987**, 30, 52–62. [Google Scholar] - Lavrenchenko, S.A. Irreducible triangulations of a torus. J. Sov. Math.
**1990**, 51, 2537–2543. [Google Scholar] [CrossRef] - Lawrencenko, S. Polyhedral suspensions of arbitrary genus. Graphs Comb.
**2010**, 26, 537–548. [Google Scholar] [CrossRef] - Lawrencenko, S. Explicit Lists of All Automorphisms of the Irreducible Toroidal Triangulations and of All Toroidal Embeddings of Their Labeled Graphs; Report Deposited at UkrNIINTI (Ukrainian Scientific Research Institute of Scientific and Technical Information); Report No. 2779-Uk87 (1 October 1987); Yangel Kharkiv Institute of Radio Electronics: Kharkiv, Ukraine, 1987. [Google Scholar]
- Harary, F. Graph Theory; Addison-Wesley: Reading, MA, USA, 1969. [Google Scholar]
- Rosen, K.H. Discrete Mathematics and Its Applications, 4th ed.; McGraw-Hill: Boston, MA, USA, 2002. [Google Scholar]
- Boas, R.P., Jr. Inequalities for the derivatives of polynomials. Math. Mag.
**1969**, 42, 165–174. [Google Scholar] [CrossRef] - White, A.T. Graphs, Groups and Surfaces. North-Holland Mathematics Studies, No. 8; North-Holland Publishing Co.: Amsterdam, The Netherlands; London, UK; American Elsevier Publishing Co., Inc.: New York, NY, USA, 1973. [Google Scholar]
- Lavrenchenko, S.A. All self-complementary simplicial 2-complexes homeomorphic to the torus or the projective plane. In Proceedings of the Baku International Topological Conference, Baku, Azerbaijan, 3–9 October 1987; p. 159. [Google Scholar]

**Figure 2.**Triangulation $T\left(G\right)$ of the torus with the graph $G={K}_{2,2,2,2}$ whose vertices are labeled with $1,\dots ,8$.

**Figure 4.**(

**a**) All vertex labelings of the tree ${T}_{1}$. (

**b**) All vertex labelings of the tree ${T}_{2}$.

**Figure 8.**(

**a**) The images in the left-hand sides of the frames correspond to: id and ${\gamma}_{\mathrm{rot}}$, respectively (upper row), and ${\gamma}_{\mathrm{ref}}$ and ${\gamma}_{\mathrm{rot}}{\gamma}_{\mathrm{ref}}$, respectively (lower row). (

**b**) Pairwise different labeled triangulations: Series 1: $T\left(G\right)$ and ${\gamma}_{\mathrm{rot}}\xb7T\left(G\right)$, respectively (upper row), ${\gamma}_{\mathrm{ref}}\xb7T\left(G\right)$ and ${\gamma}_{\mathrm{rot}}{\gamma}_{\mathrm{ref}}\xb7T\left(G\right)$, respectively (lower row).

**Figure 9.**Pairwise different labeled triangulations: Series 2: $\left(i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j\right)(-\phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}j)\xb7T\left(G\right)$ and $\left(i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j\right)(-\phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}j)\xb7{\gamma}_{\mathrm{rot}}\xb7T\left(G\right)$, respectively (upper row), $\left(i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j\right)(-\phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}j)\xb7{\gamma}_{\mathrm{ref}}\xb7T\left(G\right)$ and $\left(i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}j\right)(-\phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}j)\xb7{\gamma}_{\mathrm{rot}}{\gamma}_{\mathrm{ref}}\xb7T\left(G\right)$, respectively (lower row).

**Figure 10.**Pairwise different labeled triangulations: Series 3: $\left(i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k\right)(-\phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}k)\xb7T\left(G\right)$ and $\left(i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k\right)(-\phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}k)\xb7{\gamma}_{\mathrm{rot}}\xb7T\left(G\right)$, respectively (upper row), $\left(i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k\right)(-\phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}k)\xb7{\gamma}_{\mathrm{ref}}\xb7T\left(G\right)$ and $\left(i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}k\right)(-\phantom{\rule{-0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}k)\xb7{\gamma}_{\mathrm{rot}}{\gamma}_{\mathrm{ref}}\xb7T\left(G\right)$, respectively (lower row).

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Lawrencenko, S.; Magomedov, A.M.
Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph *K*_{2,2,2,2}. *Symmetry* **2021**, *13*, 1418.
https://doi.org/10.3390/sym13081418

**AMA Style**

Lawrencenko S, Magomedov AM.
Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph *K*_{2,2,2,2}. *Symmetry*. 2021; 13(8):1418.
https://doi.org/10.3390/sym13081418

**Chicago/Turabian Style**

Lawrencenko, Serge, and Abdulkarim M. Magomedov.
2021. "Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph *K*_{2,2,2,2}" *Symmetry* 13, no. 8: 1418.
https://doi.org/10.3390/sym13081418