Related papers: Coloring distance graphs on the plane

A graph is said to be interval colourable if it admits a proper edge-colouring using palette $\mathbb{N}$ in which the set of colours incident to each vertex is an interval. The interval colouring thickness of a graph $G$ is the minimum $k$…

The famous Erd\H{o}s distinct distances problem asks the following: how many distinct distances must exist between a set of $n$ points in the plane? There are many generalisations of this question that ask one to consider different spaces…

We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying…

Hadwiger and Haj\'{o}s conjectured that for every positive integer $t$, $K_{t+1}$-minor free graphs and $K_{t+1}$-topological minor free graphs are properly $t$-colorable, respectively. Clustered coloring version of these two conjectures…

A b-coloring of a graph is a proper coloring of its vertices such that each color class contains a vertex adjacent to at least one vertex of every other color class. The b-chromatic number of a graph is the largest integer k such that the…

For $d > 0$, define $G(\mathbb{Q}^3, d)$ to be the graph whose set of vertices is the rational space $\mathbb{Q}^3$, where two vertices are adjacent if and only if they are a Euclidean distance $d$ apart. Let $\chi(\mathbb{Q}^3, d)$ be the…

Let $\mathcal{C}_k(n)$ be the family of all connected $k$-chromatic graphs of order $n$. Given a natural number $x\geq k$, we consider the problem of finding the maximum number of $x$-colorings among graphs in $\mathcal{C}_k(n)$. When…

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this paper, we prove that for every connected graph with maximum degree at most three and…

There exists a variety of coloring problems for plane graphs, involving vertices, edges, and faces in all possible combinations. For instance, in the \emph{entire coloring} of a plane graph we are to color these three sets so that any pair…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex…

Let $G$ be an edge-colored connected graph. A path $P$ in $G$ is called a distance $\ell$-proper path if no two edges of the same color appear with fewer than $\ell$ edges in between on $P$. The graph $G$ is called $(k,\ell)$-proper…

We study weighted edge coloring of graphs, where we are given an undirected edge-weighted general multi-graph $G := (V, E)$ with weights $w : E \rightarrow [0, 1]$. The goal is to find a proper weighted coloring of the edges with as few…

We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99],…

In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings…

An edge-coloring of a graph $G$ with colors $1,...,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

An edge-coloring of a graph $G$ with colors $1,\ldots,t$ is an interval $t$-coloring if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval…

$(1^a, 2^b)$-coloring is the problem of partitioning the vertex set of a graph into $a$ independent sets and $b$ 2-independent sets. This problem was recently introduced by Choi and Liu. We study the computational complexity and extremal…

Let $P$ be a set of $n\geq 4$ points in general position in the plane. Consider all the closed straight line segments with both endpoints in $P$. Suppose that these segments are colored with the rule that disjoint segments receive different…

An edge-coloring of a graph $G$ with colors $1,...,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In 1991 Erd\H{o}s…

We call a proper edge coloring of a graph $G$ a B-coloring if every 4-cycle of $G$ is colored with four different colors. Let $q_B(G)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on…