Related papers: Paradoxical decompositions and finitary colouring …
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…
We say that a vertex-coloring of a graph is a proper k-distance domatic coloring if for each color, every vertex is within distance k from a vertex receiving that color. The maximum number of colors for which such a coloring exists is…
The better title is "Yet another FALSE proof of the 4-colour theorem." Please consider all versions of this paper as historical material on the way to a non-computer proof of the 4-colour theorem. Interpreted as proofs, all versions are…
We prove that any finite set of half-planes can be colored by two colors so that every point of the plane, which belongs to at least three half-planes in the set, is covered by half-planes of both colors. This settles a problem of Keszegh.
A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and…
The color-flavor transformation is a mathematical result that has applications to problems as diverse as lattice gauge theory, random network models, and dynamical systems. Several variants are described, and an outline of the proof is…
In a proper edge-coloring the edges of every color form a matching. A matching is induced if the end-vertices of its edges induce a matching. A strong edge-coloring is an edge-coloring in which the edges of every color form an induced…
A partition $(V_1,\ldots,V_k)$ of the vertex set of a graph $G$ with a (not necessarily proper) colouring $c$ is colourful if no two vertices in any $V_i$ have the same colour and every set $V_i$ induces a connected graph. The COLOURFUL…
A vertex colouring of some graph is called perfect if each vertex of colour $i$ has the same number $a_{ij}$ of neighbours of colour $j$. Here we determine all perfect colourings of the edge graphs of the hypercube in dimensions 4 and 5 by…
This chapter presents an introduction to graph colouring algorithms. The focus is on vertex-colouring algorithms that work for general classes of graphs with worst-case performance guarantees in a sequential model of computation. The…
In this work we reexamine the EPR paradox for composite systems with a finite number of levels. The analysis emphasizes the connection between measurements and conditional probabilities. This connection implies that when a measurement is…
Peculiar measurements can be obtained on systems that undergo both pre- and post-selection. We prove a conjecture from [1] on logical Pre- and Post-Selection (PPS) paradoxes for a restricted case. We prove that all of these paradoxes admit…
We give a very short self-contained combinatorial proof of the Babson-Kozlov conjecture, by presenting a cochain whose coboundary is the desired power of the characteristic class.
The Sudoku number has been defined under various names, indicating it is a natural concept. There are four variants of this parameter, that can be related to the maximum and minimum size of a critical set in a graph colouring problem. For…
Given an edge colouring of a graph with a set of $m$ colours, we say that the graph is (exactly) $m$-coloured if each of the colours is used. We consider edge colourings of the complete graph on $\mathbb{N}$ with infinitely many colours and…
We make an attempt at proving the Four Colour Theorem in six pages.
We develop a probabilistic technique for colorizing grayscale natural images. In light of the intrinsic uncertainty of this task, the proposed probabilistic framework has numerous desirable properties. In particular, our model is able to…
We study reflection-symmetric realisations of symmetric graphs in the plane that allow a continuous symmetry and edge-length preserving deformation. To do so, we identify a necessary combinatorial condition on graphs with…
The study of a machine learning problem is in many ways is difficult to separate from the study of the loss function being used. One avenue of inquiry has been to look at these loss functions in terms of their properties as scoring rules…
We study a graph coloring problem motivated by a fun Sudoku-style puzzle. Given a bipartition of the edges of a graph into {\em near} and {\em far} sets and an integer threshold $t$, a {\em threshold-coloring} of the graph is an assignment…