Related papers: An Iterative Solution of the Three-colour Problem …
Circular coloring is a constraints satisfaction problem where colors are assigned to nodes in a graph in such a way that every pair of connected nodes has two consecutive colors (the first color being consecutive to the last). We study…
Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times…
Recently, connections have been explored between the complexity of finite problems in graph theory and the complexity of their infinite counterparts. As is shown in our paper (and in independent work of Tirza Hirst and D. Harel from a…
The use of algebraic techniques to solve combinatorial problems is studied in this paper. We formulate the rainbow connectivity problem as a system of polynomial equations. We first consider the case of two colors for which the problem is…
Usual approaches for image recoloring, such as local filtering by transfer functions and global histogram remapping, lack of accurate control or miss small groups of important pixels. In this paper, we introduce a triangle-based structuring…
We study Kaczmarz type methods to solve consistent linear matrix equations. We first present a block Kaczmarz (BK) method that employs a deterministic cyclic row selection strategy. Assuming that the associated coefficient matrix has full…
A multi-component electron model on a lattice is constructed whose ground state exhibits a spontaneous ordering which follows the rule of map-coloring used in the solution of the four color problem. The number of components is determined by…
The list coloring problem is a variation of the classical vertex coloring problem, extensively studied in recent years, where each vertex has a restricted list of allowed colors, and having some variations as the $(\gamma,\mu)$-coloring,…
We study a certain relaxation of the classic vertex coloring problem, namely, a coloring of vertices of undirected, simple graphs, such that there are no monochromatic triangles. We give the first classification of the problem in terms of…
We address the six vertex model on a random lattice, which in combinatorial terms corresponds to the enumeration of weighted 4-valent planar maps equipped with an Eulerian orientation. This problem was exactly, albeit non-rigorously solved…
We consider solutions to the $4$-color problem for the vertices of sphere triangulations with degree sequence $6,...,6,4,4,4,4,4,4$. We sort these solutions into combinatorial types and show that each generic type $\tau$ is parametrized by…
We revisit the problem of hard particles on planar random tetravalent graphs in view of recent combinatorial techniques relating planar diagrams to decorated trees. We show how to recover the two-matrix model solution to this problem in…
We define and solve classes of sparse matrix problems that arise in multilevel modeling and data analysis. The classes are indexed by the number of nested units, with two-level problems corresponding to the common situation in which data on…
We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element $i$ in each color class there exists a triple which does not contain $i$. We give a…
We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets. With this we give, for any fixed $k\ge 3$ and $\varepsilon>0$, a randomised…
The proof uses the property that the vertices of a triangulated planar graph can be four coloured if the triangles can have a +1 or -1 orientation so that the sum of the triangle orientations around each vertex is a multiple of 3. Such…
A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions…
We show how to couple phase-oscillators on a graph so that collective dynamics "searches" for the coloring of that graph as it relaxes toward the dynamical equilibrium. This translates a combinatorial optimization problem (graph coloring)…
After a brief discussion of the history of the problem, we propose a generalization of the map colouring problem to higher dimensions.
We are interested in the independence number of large random simply generated trees and related parameters, such as their matching number or the kernel dimension of their adjacency matrix. We express these quantities using a canonical…