Related papers: Symmetrization for finitely dependent colouring

In this paper we define the chromatic number of a lattice: It is the least number of colors one needs to color the interiors of the cells of the Voronoi tessellation of a lattice so that no two cells sharing a facet are of the same color.…

We prove that the vertices of every $(r + 1)$-uniform hypergraph with maximum degree $\Delta$ may be coloured with $c(\frac{\Delta}{d + 1})^{1/r}$ colours such that each vertex is in at most $d$ monochromatic edges. This result, which is…

We show that, for every $\epsilon>0$, the 4-regular tree has an fiid 4-coloring where a given vertex is assigned the 4th color with probability at most $\epsilon$. We also construct 5-colorings of $T_6$ improving known bounds on the…

Tiling models are classical statistical models in which different geometric shapes, the tiles, are packed together such that they cover space completely. In this paper we discuss a class of two-dimensional tiling models in which the tiles…

Let G be a combinatorial graph with vertices V and edges E. A proper coloring of G is an assignment of colors to the vertices such that no edge connects two vertices of the same color. These are the colorings considered in the famous Four…

We study the list-chromatic number and the coloring number of graphs, especially uncountable graphs. We show that the coloring number of a graph coincides with its list-chromatic number provided that the diamond principle holds. Under the…

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $\Delta$ and $\pi$. The ``approximate'' version states that, for any Borel probability…

A new realization of the conformal algebra is studied which mimics the behaviour of a statistical system on a discrete albeit infinite lattice. The two-point function is found from the requirement that it transforms covariantly under this…

We show that any finite-entropy, countable-valued finitary factor of an i.i.d process can also be expressed as a finitary factor of a finite-valued i.i.d process whose entropy is arbitrarily close to the target process. As an application,…

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…

It is argued that a dual symmetry is needed to naturally explain experimental limits on color confinement. Since color is an exact symmetry the only possibility is that this symmetry be a dual symmetry, related to non trivial spatial…

Given a set D of positive integers, the associated distance graph on the integers is the graph with the integers as vertices and an edge between distinct vertices if their difference lies in D. We investigate the chromatic numbers of…

The locating chromatic number of a graph is the smallest integer $n$ such that there is a proper $n$-coloring $c$ and every vertex has a unique vector of distances to colors in $c$. We explore the necessary conditions and provide sufficient…

Given a multigraph $G$ and a positive integer $t$, the distance-$t$ chromatic index of $G$ is the least number of colours needed for a colouring of the edges so that every pair of distinct edges connected by a path of fewer than $t$ edges…

This paper presents three main results on coloring discrete $d$-pseudomanifolds: $(1)$ the general chromatic bounds $d+1 \leq X(K) \leq 2d+2$ for any $d$-pseudomanifold $K$; $(2)$ an improved bound $X(K) \leq 2d+1$ for pseudomanifolds…

Colouring problems arising from group-based constructions provide a natural link between combinatorics and algebra, particularly in the study of Cayley graphs and Latin squares. We introduce the notion of colouring bijections of finite…

We show that all balanced d-lattices must be complemented, answering a question of Chajda and Eigenthaler. (A bounded lattice is balanced if any two congruences agree on their 1-classes iff they agree on their 0-classes.) Our main tool is…

In a colouring of $\mathbb{R}^d$ a pair $(S,s_0)$ with $S\subseteq \mathbb{R}^d$ and with $s_0\in S$ is \emph{almost monochromatic} if $S\setminus \{s_0\}$ is monochromatic but $S$ is not. We consider questions about finding almost…

We give new positive results on the long-standing open problem of geometric covering decomposition for homothetic polygons. In particular, we prove that for any positive integer k, every finite set of points in R^3 can be colored with k…

We study a new variant of \emph{connected coloring} of graphs based on the concept of \emph{strong} edge coloring (every color class forms an \emph{induced} matching). In particular, an edge-colored path is \emph{strongly proper} if its…