Related papers: Application of Vertex coloring in a particular tri…
A $b$-coloring of a graph is a proper vertex coloring such that each color class contains a vertex that sees all other colors in its neighborhood. The $b$-coloring problem, in which the task is to decide whether a graph admits a…
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 introduce and solve a two-matrix model for the tri-coloring problem of the vertices of a random triangulation. We present three different solutions: (i) by orthogonal polynomial techniques (ii) by use of a discrete Hirota bilinear…
In this paper, we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. In previous work, we gave a polynomial…
In this paper, we study Dehn colorings for spatial graphs, and give a family of spatial graph invariants that are called vertex-weight invariants. We give some examples of spatial graphs that can be distinguished by a vertex-weight…
A $k$-{\it edge-weighting} $w$ of a graph $G$ is an assignment of an integer weight, $w(e)\in \{1,\dots, k\}$, to each edge $e$. An edge weighting naturally induces a vertex coloring $c$ by defining $c(u)=\sum_{u\sim e} w(e)$ for every $u…
We study a new variant of graph coloring by adding a connectivity constraint. A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph $G$ is said to be…
We study the behavior of the Douglas-Rachford algorithm on the graph vertex-coloring problem. Given a graph and a number of colors, the goal is to find a coloring of the vertices so that all adjacent vertex pairs have different colors. In…
We show that any planar drawing of a forest of three stars whose vertices are constrained to be at fixed vertex locations may require $\Omega(n^\frac{2}{3})$ edges each having $\Omega(n^\frac{1}{3})$ bends in the worst case. The lower bound…
The asymmetric coloring number of a graph is the minimum number of colors needed to color its vertices, so that no non-trivial automorphism preserves the color classes. We investigate the asymmetric coloring number of graphs that are…
A proper vertex-colouring of a simple graph $G$ is said to be odd if, for every non-isolated vertex $v$ of $G$, some colour appears an odd number of times in the neighbourhood of $v$. We show that if $G$ embeds in the torus, then it admits…
The attempts to prove the Four Color Problem last for long years. A little hope arises that the properties of the minimal partial triangulations will be very useful for the solution of the Four Color Problem. That is why the material of…
Recently, it was proved that triangle-free intersection graphs of $n$ line segments in the plane can have chromatic number as large as $\Theta(\log\log n)$. Essentially the same construction produces $\Theta(\log\log n)$-chromatic…
This article is devoted to the properties of the planar triangulations. The conjugated planar triangulation will be introduced and on the base of the properties, which were achieved by the other authors there will be proved some theorems,…
We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.
A proper vertex-colouring of a graph G is said to be locally identifying if for any pair u,v of adjacent vertices with distinct closed neighbourhoods, the sets of colours in the closed neighbourhoods of u and v are different. We show that…
An equitable tree-$k$-coloring of a graph is a vertex coloring on $k$ colors so that every color class incudes a forest and the sizes of any two color classes differ by at most one.This kind of coloring was first introduced in 2013 and can…
The chromatic index of a cubic graph is either 3 or 4. Edge-Kempe switching, which can be used to transform edge-colorings, is here considered for 3-edge-colorings of cubic graphs. Computational results for edge-Kempe switching of cubic…
We consider extensions of Brooks' classic theorem on vertex coloring where some colors cannot be used on certain vertices. In particular we prove that if $G$ is a connected graph with maximum degree $\Delta(G) \geq 4$ that is not a complete…
For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)\neq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring…