symmetry S S Article Generating the Triangulations of the Torus with the Vertex-Labeled Complete 4-Partite Graph K 2,2,2,2 Serge Lawrencenko 1, * and Abdulkarim M. Magomedov 2   Citation: 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 Academic Editor: Lorentz Jäntschi Received: 9 July 2021 Accepted: 30 July 2021 Published: 3 August 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 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/). 1 Institute of Service Technologies, Russian State University of Tourism and Service, 99 Glavnaya Street, Cherkizovo, Pushkinsky District, 141221 Moscow Region, Russia 2 Department of Discrete Mathematics and Informatics, Dagestan State University, 43-A Gadjieva, 367000 Makhachkala, Russia; magomedtagir1@yandex.ru * Correspondence: lawrencenko@hotmail.com Abstract: Using the orbit decomposition, a new enumerative polynomial P(x) is introduced for abstract (simplicial) complexes of a given type, e.g., trees with a fixed number of vertices or trian- gulations of the torus with a fixed graph. The polynomial has the following three useful properties. (I) The value P(1) is equal to the total number of unlabeled complexes (of a given type). (II) The value of the derivative P ′ (1) is equal to the total number of nontrivial automorphisms when counted across all unlabeled complexes. (III) The integral of P(x) from 0 to 1 is equal to the total number of vertex-labeled complexes, divided by the order of the acting group. The enumerative polynomial P(x) is demonstrated for trees and then is applied to the triangulations of the torus with the vertex- labeled complete four-partite graph G = K 2,2,2,2 , in which specific case P(x)= x 31 . The graph G embeds in the torus as a triangulation, T(G). The automorphism group of G naturally acts on the set of triangulations of the torus with the vertex-labeled graph G. For the first time, by a combination of algebraic and symmetry techniques, all vertex-labeled triangulations of the torus (12 in number) with the graph G are classified intelligently without using computing technology, in a uniform and systematic way. It is helpful to notice that the graph G can be converted to the Cayley graph of the quaternion group Q 8 with the three imaginary quaternions i, j, k as generators. Keywords: group action; orbit decomposition; polynomial; graph; tree; triangulation; torus; auto- morphism; quaternion group 1. Introduction Graph theory and its applications (polyhedra, enumeration, coloring, fullerenes, etc.) has received increasing attention in recent years [1–5], which has paved the way for more directions of research. In labeled graph enumeration problems, the vertices of the graph are labeled to be distinguishable from each other, while in unlabeled graph enumeration problems any admissible permutation of the vertices is regarded as producing the same graph, so the vertices are considered unlabeled. In general, labeled problems are usually easier than unlabeled ones. For example, Cayley’s tree formula [6,7] gives the number, n n−2 , of trees with n vertices bijectively labeled by 1, ... , n, whereas the number of unlabeled trees with n vertices can only be evaluated as the coefficients of a generating function [8,9]. The number n n−2 can be interpreted as the number of different ways of placing n given folders on the desktop into the one a priori chosen out of them and fixed (the root folder). The orbit decomposition [10] is an important tool for reducing unlabeled problems to labeled ones: Each unlabeled class is considered to be a symmetry class, or an isomorphism class, of labeled graphs. In the current paper we introduce a new enumerative polynomial P( x) which is a bridge between the labeled and unlabeled settings. A graph consists of a finite set of vertices, some of which are connected by edges. To “embed a graph in a surface” is, loosely speaking, to draw it on that surface without any edges crossing. An embedding of a graph in a surface is called a closed 2-cell embedding if the surface breaks up into a union of connected components, the faces of the embedding, Symmetry 2021, 13, 1418. https://doi.org/10.3390/sym13081418 https://www.mdpi.com/journal/symmetry