Related papers: Triangle-Free Triangulations, Hyperplane Arrangeme…
We consider a random graph model evolving in discrete time-steps that is based on 3-interactions among vertices. Triangles, edges and vertices have different weights; objects with larger weight are more likely to participate in future…
We study the connectedness and the diameter of orthogonality graphs of upper triangular matrix algebras over arbitrary fields.
The intersection matrix of a simplicial complex has entries equal to the rank of the intersection of its facets. In [1] the authors prove the intersection matrix is enough to determine a triangulation of a surface up to isomorphism. In this…
The visibility graph of a simple polygon represents visibility relations between its vertices. Knowing the correct order of the vertices around the boundary of a polygon and its visibility graph, it is an open problem to locate the vertices…
Let $V$ denote a nonzero finite-dimensional vector space. A tridiagonal pair on $V$ is an ordered pair $A, A^*$ of maps in ${\rm End}(V)$ such that (i) each of $A, A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i…
A fixed set of vertices in the plane may have multiple planar straight-line triangulations in which the degree of each vertex is the same. As such, the degree information does not completely determine the triangulation. We show that even if…
This article is devoted to the properties of the planar triangulations. The conjugated planar triangulation will be introduced and on the base of the properties, which were achieved by the other authors there will be proved some theorems,…
Let $X$ be a two-sided subshift on a finite alphabet endowed with a mixing probability measure which is positive on all cylinders in $X$. We show that there exist arbitrarily small finite overlapping union of shifted cylinders which…
We study the folding of the regular two-dimensional triangular lattice embedded in the regular three-dimensional Face-Centred Cubic lattice, a discrete model for the crumpling of membranes. Possible folds are complete planar folds, folds…
It is shown that every orthogonal terrain, i.e., an orthogonal (right-angled) polyhedron based on a rectangle that meets every vertical line in a segment, has a grid unfolding: its surface may be unfolded to a single non-overlapping piece…
Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of…
Diagrams enable the use of various algebraic and geometric tools for analysing and classifying knots. In this paper we introduce a new diagrammatic representation of triply periodic entangled structures (TP tangles), which are embeddings of…
We construct a continuum model of twisted trilayer graphene using {\it ab initio} density-functional-theory calculations, and apply it to address twisted trilayer electronic structure. Our model accounts for moir\'e variation in site…
In this paper, we consider domino tilings of regions of the form $\mathcal{D} \times [0,n]$, where $\mathcal{D}$ is a simply connected planar region and $n \in \mathbb{N}$. It turns out that, in nontrivial examples, the set of such tilings…
We develop tools to study the topology and geometry of self-affine fractals in dimension three and higher. We use the self-affine structure and obtain rather detailed information about the connectedness of interior and boundary sets, and on…
Consider the map $S$ which sends a planar polygon $P$ to a new polygon $S(P)$ whose vertices are the intersection points of second nearest sides of $P$. This map is the inverse of the famous pentagram map. In this paper we investigate the…
Three--dimensional colored triangulations are gluings of tetrahedra whose faces carry the colors 0, 1, 2, 3 and in which the attaching maps between tetrahedra are defined using the colors. This framework makes it possible to generalize the…
I study the interplay of errors with transversal gates in 3D color codes, and introduce some new such gates. Two features of the transversal T gate stand out: (i) it naturally defines a set of correctable errors, and (ii) it exhibits a…
Consider a periodical (in two independent directions) tiling of the plane with polygons (faces). In this article we shall only give examples using squares, regular hexagons, equilateral triangles and parallelograms ("unions" of two…
Triangulation graph staining is sufficient for planar graph staining. This article will focus on triangulation and the nature of the color change channel of the staining tool. By construction, the four colors of the vertex are converted…