Related papers: Counting gluings of octahedra
Esperet and Joret proved that planar graphs with bounded maximum degree are 3-colorable with bounded clustering. Liu and Wood asked whether the conclusion holds with the assumption of the bounded maximum degree replaced by assuming that no…
We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 "templates" (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not…
A degree-regular triangulation is one in which each vertex has identical degree. Our main result is that any such triangulation of a (possibly non-compact) surface $S$ is geometric, that is, it is combinatorially equivalent to a geodesic…
We prove elegant trilinear formulas connecting products of volumes of Euclidean tetrahedra with vertices taken from a given set of 6 points. We propose a way for generalizing those formulas.
We study a three-dimensional analogue to the well-known graph visualization approach known as arc diagrams. We provide several algorithms that achieve good angular resolution for 3D arc diagrams, even for cases when the arcs must project to…
A plane graph is l-facially k-colourable if its vertices can be coloured with k colours such that any two distinct vertices on a facial segment of length at most l are coloured differently. We prove that every plane graph is 3-facially…
The triangulation complexity of a closed orientable 3-manifold is the minimal number of tetrahedra in any triangulation of the manifold. The main theorem of the paper gives upper and lower bounds on the triangulation complexity of any…
An $\ell$-facial edge-coloring of a plane graph is a coloring of its edges such that any two edges at distance at most $\ell$ on a boundary walk of any face receive distinct colors. It is the edge-coloring variant of the $\ell$-facial…
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,…
We consider the problem of counting the number of 3-colourings of the edges (bonds) of the 4-8 lattice and the 3-12 lattice. These lattices are Archimedean with coordination number 3, and can be regarded as decorated versions of the square…
For any cluster algebra whose underlying combinatorial data can be encoded by a bordered surface with marked points, we construct a geometric realization in terms of suitable decorated Teichmueller space of the surface. On the geometric…
We study the connectedness and the diameter of orthogonality graphs of upper triangular matrix algebras over arbitrary fields.
We propose an algorithm to create a 3-colorable Delaunay Triangulation. The input of the problem we are trying to solve is a set X of n twodimensional points. The output is a 3-colorable two-dimensional Delaunay triangulation T for X U Y ,…
The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, \Sigma)$…
In 1996, Youngs proved that any quadrangulation of the real projective plane is not 3-chromatic. This result has been extended in various directions over the years, including to other non-orientable closed surfaces, higher-dimensional…
We study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations consisting of a single line and three spheres for which there are infinitely many lines tangent to the three spheres that also…
A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…
Planar bipartite graphs can be represented as touching graphs of horizontal and vertical segments in $\mathbb{R}^2$. We study a generalization in space: touching graphs of axis-aligned rectangles in $\mathbb{R}^3$, and prove that planar…
A triangulation of a punctured or pinched surface is irreducible if no edge can be shrunk without producing multiple edges or changing the topological type of the surface. The finiteness of the set of (non-isomorphic) irreducible…
We develop a general method for constructing random manifolds and submanifolds in arbitrary dimensions. The method is based on associating colors to the vertices of a triangulated manifold, as in recent work for curves in 3-dimensional…