graph colouring - meaning and definition. What is graph colouring
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What (who) is graph colouring - definition

ASSIGNMENT OF COLORS TO ELEMENTS OF A GRAPH SUBJECT TO CERTAIN CONSTRAINTS
Colouring algorithm; Coloring algorithm; Graph coloring algorithm; Chromatic number; Graph colouring problems; Graph coloring problem; Colored graph; Graph Colouring; Vertex chromatic number; K-vertex colorable; Vertex color; Graph colouring problem; Graph colouring; Three-Colorable Graph; Three-colorable graph; Vertex-colouring; Vertex colouring; Vertex coloring; Coloring problem; Colouring problem; Two-colorable graph; Graph two-coloring; Graph Two-Coloring; Graph coloration; Graph color; K-colouring; 3-colourability; Colourability; Proper coloring; K-coloring; Network coloring; Network colouring; K-chromatic graph; Distributed graph coloring; Cole–Vishkin algorithm; Cole-Vishkin algorithm; Mycielski's theorem; K-colorable; Unlabeled coloring; Vector chromatic number; Face coloring; Algorithms for graph coloring; Parallel algorithms for graph coloring; Applications of graph coloring; Decentralized graph coloring; Computational complexity of graph coloring
  • 3}}}} (blue) admits a 3-coloring; the other graphs admit a 2-coloring.
  • This graph can be 3-colored in 12 different ways.
  • Two greedy colorings of the same graph using different vertex orders. The right example generalizes to 2-colorable graphs with ''n'' vertices, where the greedy algorithm expends <math>n/2</math> colors.
  • A proper vertex coloring of the [[Petersen graph]] with 3 colors, the minimum number possible.

graph colouring         
<application> A constraint-satisfaction problem often used as a test case in research, which also turns out to be equivalent to certain real-world problems (e.g. {register allocation}). Given a connected graph and a fixed number of colours, the problem is to assign a colour to each node, subject to the constraint that any two connected nodes cannot be assigned the same colour. This is an example of an NP-complete problem. See also four colour map theorem.
graph coloring         
chromatic number         

Wikipedia

Graph coloring

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is partly pedagogical, and partly because some problems are best studied in their non-vertex form, as in the case of edge coloring.

The convention of using colors originates from coloring the countries of a map, where each face is literally colored. This was generalized to coloring the faces of a graph embedded in the plane. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. In mathematical and computer representations, it is typical to use the first few positive or non-negative integers as the "colors". In general, one can use any finite set as the "color set". The nature of the coloring problem depends on the number of colors but not on what they are.

Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku. Graph coloring is still a very active field of research.

Note: Many terms used in this article are defined in Glossary of graph theory.