相关论文: Symmetry and colorings: Some results and open prob…
We survey some principal results and open problems related to colorings of geometric and algebraic objects endowed with symmetries, concentrating the exposition on the maximal symmetry numbers of such objects.
We obtain tilings with a singular point by applying conformal maps on regular tilings of the Euclidean plane, and determine its symmetries. The resulting tilings are then symmetrically colored by applying the same conformal maps on…
Some symmetry problems are formulated and solved. New simple proofs are given for the earlier studied symmetry problems.
Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful…
Some basic features of confinement are reviewed, in particular the symmetry patterns of the dual dynamics. Open problems and possible directions of progress are discussed.
Color symmetry is an extension of symmetry imposed by isometric transformations and means that the colors of geometrical objects are assigned according to the symmetry properties of the objects. A color symmetry permutes the coloring of the…
Partially or totally unsolved questions in number theory and geometry especially, such as coloration problems, elementary geometric conjectures, partitions, generalized periods of a number, length of a generalized period, arithmetic and…
Symmetries and reductions of some algebraic equations are considered. Transformations that preserve the form of several algebraic equations, as well as transformations that reduce the degree of these equations, are described. Illustrative…
We address the problem of finding harmonic colors, this problem has many applications, from fashion to industrial design. In order to solve this problem we consider that colors follow normal distributions in tone (chroma and lightness) and…
A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of…
We use some fundamental ideas from complex analysis to create symmetric images and animations. Using a domain coloring algorithm, we generate mappings to the entire complex plane or the hyperbolic upper half-plane. The resulting designs can…
We consider triangulations of surfaces with edges painted three colors so that edges of each triangle have different colors. Such structures arise as Belyi data (or Grothendieck dessins d'enfant), on the other hand they enumerate pairs of…
This paper describes several new problems and ideas concerning algebraic geometry and complexity theory. It first uses the idea of coloring graphs with elements of finite fields. This procedure then shows that graph coloring problems can be…
At a first glance, the problem of illuminating the boundary of a convex body by external light sources and the problem of covering a convex body by its smaller positive homothetic copies appear to be quite different. They are in fact two…
The study of symmetric configurations $v_3$ with block size 3 has a long and rich history. In this paper we consider two colouring problems which arise naturally in the study of these structures. The first of these is weak colouring, in…
We prove lower and upper bounds for the chromatic number of certain hypergraphs defined by geometric regions. This problem has close relations to conflict-free colorings. One of the most interesting type of regions to consider for this…
After a brief discussion of the history of the problem, we propose a generalization of the map colouring problem to higher dimensions.
The main aim of the paper is to study in greater detail absolutely homogeneous structures (that is, objects with the property that each partial isomorphism extends to a global automorphism), with special emphasis on metric spaces and…
We investigate the complexity of generalizations of colourings (acyclic colourings, $(k,\ell)$-colourings, homomorphisms, and matrix partitions), for the class of transitive digraphs. Even though transitive digraphs are nicely structured,…
A general (convex) polytope $P\subset\mathbb R^d$ and its edge-graph $G_P$ can have very distinct symmetry properties. We construct a coloring (of the vertices and edges) of the edge-graph so that the combinatorial symmetry group of the…