Related papers: Chromatic numbers for the hyperbolic plane and dis…

Quantum chromodynamics (QCD) with a general number of colors, $\Nc$, provides a powerful theoretical laboratory to explore the dynamics of non-Abelian gauge theories. Although $\Nc =3$ does not look a large number, the $1/\Nc$ expansion…

Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring…

If you color a table using k colors, and throw a needle randomly on it, for some proper definition, you get a certain probability that the endpoints will fall on different colors. How can one make this probability maximal? This problem is…

We study the unit distance and distinct distances problems over the planar hypercomplex numbers: the dual numbers $\mathbb{D}$ and the double numbers $\mathbb{S}$. We show that the distinct distances problem in $\mathbb{S}^2$ behaves…

We solve the problem of finding a sharp upper bound on the minimum angle formed by $N$ points in the Euclidean and Hyperbolic planes.

A $t$-tone $k$-coloring of a graph $G$ assigns a set of $t$ distinct colors from $\{1, \dots, k\}$ to each vertex so that vertices at distance $d$ share fewer than $d$ common colors. The $t$-tone chromatic number of $G$ is the minimum $k$…

We give a new, simple proof for the lower bound of the chromatic number of the Euclidean plane with two forbidden distances, based on a graph with only 16 vertices.

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more…

In this paper, we consider the colorful $k$-center problem, which is a generalization of the well-known $k$-center problem. Here, we are given red and blue points in a metric space, and a coverage requirement for each color. The goal is to…

Let $G$ be a simple finite connected graph of order $n$. The detour distance between two distinct vertices $u$ and $v$ denoted by $D(u,v)$ is the length of a longest $uv$-path in $G$. A hamiltonian coloring $h$ of a graph $G$ of order $n$…

We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99],…

We introduce a variant of the vertex-distinguishing edge coloring problem, where each edge is assigned a subset of colors. The label of a vertex is the union of the sets of colors on edges incident to it. In this paper we investigate the…

We study the problem of consistent and homogeneous colourings for increasing families of dyadic intervals. We determine when this problem can be solved and when not.

Many variations of the classical graph coloring model have been intensively studied due to their multiple applications; scheduling problems and aircraft assignments, for instance, motivate the robust coloring problem. This model gets to…

While every plane triangulation is colourable with three or four colours, Heawood showed that a plane triangulation is 3-colourable if and only if every vertex has even degree. In $d \geq 3$ dimensions, however, every $k \geq d+1$ may occur…

The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a…

This paper extends the scenario of the Four Color Theorem in the following way. Let H(d,k) be the set of all k-uniform hypergraphs that can be (linearly) embedded into R^d. We investigate lower and upper bounds on the maximum (weak and…

A colouring of a hypergraph's vertices is polychromatic if every hyperedge contains at least one vertex of each colour; the polychromatic number is the maximum number of colours in such a colouring. Its dual, the cover-decomposition number,…

Fix $d\ge2$ and consider a uniformly random set $P$ of $n$ points in $[0,1]^{d}$. Let $G$ be the Hasse diagram of $P$ (with respect to the coordinatewise partial order), or alternatively let $G$ be the Delaunay graph of $P$ with respect to…

We show that if a coloring of the plane has the properties that any two points at distance one are colored differently and the plane is partitioned into uniformly colored triangles under certain conditions, then it requires at least seven…