Related papers: A characterisation of 3-colourable 3-dimensional t…
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…
In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed,…
We prove that for every graph $G$, given fixed locations for the vertices of $G$ in $\mathbb{Z}^3$, there is a three-dimensional grid-drawing of $G$ with one bend per edge. The best previous bound was three bends per edge.
Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A short survey, preprint 2013] writes, without any proof, that an oriented cycle $\vec C$ can be colored with three colors if and only if $\lambda(\vec C)=0$, where…
We prove that a wide range of coloring problems in graphs on surfaces can be resolved by inspecting a finite number of configurations.
DeVos and Seymour (2003) proved that for every set $C$ of 3-colorings of a set $X$ of vertices, there exists a plane graph $G$ with vertices of $X$ incident with the outer face such that a 3-coloring of $X$ extends to a 3-coloring of $G$ if…
We discuss the existence of the angle between two curves in Teichm\"uller spaces and show that, in any infinite dimensional Teichm\"uller space, there exist infinitely many geodesic triangles each of which has the same three vertices and…
We consider edge colorings of a graph in such a way that each two different triangles have distinct colorings. It is an extension of the well-known idea of distinguishing all maximal stars in a graph. It was introduced in literature in 1985…
Consider the graph $\mathbb{H}(d)$ whose vertex set is the hyperbolic plane, where two points are connected with an edge when their distance is equal to some $d>0$. Asking for the chromatic number of this graph is the hyperbolic analogue to…
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…
The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge-coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by K\H{o}nig, $d$ colors suffice if the graph is bipartite. We investigate…
The paper deals with an extremal problem concerning colorings of hypergraphs with bounded edge degrees. Consider the family of $b$-simple hypergraphs, in which any two edges do not share more than $b$ common vertices. We prove that for…
We show that if a closed $C^1$-smooth surface in a Riemannian manifold has bounded Kolasinski--Menger energy, then it can be triangulated with triangles whose number is bounded by the energy and the area. Each of the triangles is an image…
We prove a variation on Heawood list-coloring for graphs on surfaces, modeled on Thomassen's planar 5-list-coloring theorem. For epsilon>0 define the Heawood number to be H(epsilon)=Floor((7+Sqrt[24*epsilon+1])/2). We prove that, except for…
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…
We consider the Hadwiger-Nelson problem on the chromatic number of the plane under conditions of coloring a map containing a finite number of vertices in any bounded region. Woodall (1973) and Townsend (1981) showed that at least 6 colors…
We prove that the following problem is co-RE-complete and thus undecidable: given three simple polygons, is there a tiling of the plane where every tile is an isometry of one of the three polygons (either allowing or forbidding…
A properly colored cycle (path) in an edge-colored graph is a cycle (path) with consecutive edges assigned distinct colors. A monochromatic triangle is a cycle of length $3$ with the edges assigned a same color. It is known that every…
We conjecture that every graph of minimum degree five with no separating triangles and drawn in the plane with one crossing is 4-colorable. In this paper, we use computer enumeration to show that this conjecture holds for all graphs with at…
For every fixed integer $k \geq 1$, we prove that $k$-Edge Colouring is fixed-parameter-tractable when parameterized by the number of vertices of maximum degree.