Related papers: Colouring isonemal fabrics with more than two colo…
Perfect colouring of isonemal fabrics by thin striping of warp and weft with more than two colours is examined. Examples of thin striping in all possible species with no redundancy and with redundant cells arranged as twills are given.…
Perfect colouring of isonemal fabrics by thin and thick striping of warp and weft with more than two colours is introduced. Conditions that prevent perfect colouring by striping are derived, and it is shown that avoiding them is sufficient…
Perfect colouring of isonemal fabrics by thick striping of warp and weft and the closely related topic of isonemal prefabrics that fall apart are reconsidered and their relation further explored. The catalogue of isonemal prefabrics of…
Perfect colouring of isonemal fabrics by thin striping of warp and weft and the closely related topic of isonemal prefabrics that fall apart are reconsidered and their relation further explored. The catalogue of isonemal prefabrics that…
This paper continues the refinement of Richard Roth's taxonomy of isonemal weaving designs through types 11--32 of the 39 in order to solve three problems for those designs: which designs exist in various sizes, which prefabrics can be…
This paper refines Richard Roth's taxonomy of isonemal weaving designs through the final types 33--39 in order to complete the solution of three problems for those designs: which designs exist in various sizes, which prefabrics can be…
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…
Arrangements of pseudolines are a widely studied generalization of line arrangements. They are defined as a finite family of infinite curves in the Euclidean plane, any two of which intersect at exactly one point. One can state various…
Irredundance coloring of $G$ is a proper coloring in which there exists a maximal irredundant set $R$ such that all the vertices of $R$ have different colors. The minimum number of colors required for an irredundance coloring of $G$ is…
Isonemal weaving designs, introduced into mathematical literature by Gr\"unbaum and Shephard, were classified into thirty-nine infinite sets and a small number of exceptions by Richard Roth. This paper refines Roth's taxonomy for the first…
Two colourings of a graph are orthogonal if they have the property that when two vertices are coloured with the same colour in one colouring, then those vertices receive distinct colours in the other colouring. In this paper, orthogonal…
Triangulation graph staining is sufficient for planar graph staining. This article will focus on triangulation and the nature of the color change channel of the staining tool. By construction, the four colors of the vertex are converted…
Suppose that the vertices of ${\mathbb Z}^d$ are assigned random colors via a finitary factor of independent identically distributed (iid) vertex-labels. That is, the color of vertex $v$ is determined by a rule that examines the labels…
A textile fabric consists of countless parallel vertical yarns (warps) and horizontal yarns (wefts). While common looms can weave repetitive patterns, Jacquard looms can weave the patterns without repetition restrictions. A pattern in which…
We present a general framework for constructing error-correcting codes using distributed graph coloring under the LOCAL model. Building on the correspondence between independent sets in the confusion graph and valid codes, we show that the…
We prove the existence of a finitely dependent proper colouring of the integer lattice Z^d that is fully isometry-invariant in law, for all dimensions d. Previously this was known only for d=1, while only translation-invariant examples were…
We say that a vertex or edge colouring of a graph is distinguishing if the only automorphism that preserves this colouring is the identity. A (proper) distinguishing colouring is irreducible if there is no possibility of merging two…
Consider the plane as a checkerboard, with each unit square colored black or white in an arbitrary manner. In a previous paper we showed that for any such coloring there are straight line segments, of arbitrarily large length, such that the…
A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible…
For positive integers $n$ and $r$, we consider $n$-vertex graphs with the maximum number of $r$-edge-colorings with no copy of a triangle where exactly two colors appear. We prove that, if $2 \leq r \leq 26$ and $n$ is sufficiently large,…