Computational problems in graph theory | Graph theory | Extensions and generalizations of graphs | NP-complete problems | NP-hard problems | 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. (Wikipedia).
Edge Colorings and Chromatic Index of Graphs | Graph Theory
We introduce edge colorings of graphs and the edge chromatic number of graphs, also called the chromatic index. We'll talk about k-colorings/k-edge colorings, minimum edge colorings, edge colourings as matchings, edge colourings as functions, and see examples and non-examples of edge color
From playlist Graph Theory
Introduction to Vertex Coloring and the Chromatic Number of a Graph
This video introduces vertex coloring and provides example of how to determine the chromatic number of a graph. mathispower4u.com
From playlist Graph Theory (Discrete Math)
Vertex Colorings and the Chromatic Number of Graphs | Graph Theory
What is a proper vertex coloring of a graph? We'll be introducing graph colorings with examples and related definitions in today's graph theory video lesson! A proper coloring (or just: coloring) of a graph, G, is an assignment of colors (or, more generally, labels) to the vertices of G s
From playlist Graph Theory
Discrete Math II - 10.8.1 Graph Coloring
This video focuses on graph coloring, in which color the vertices of a graph so that no two adjacent vertices have the same color. Most often, graph coloring is used for scheduling purposes, as we can determine when there are conflicts in scheduling if two vertices are the same color. Vi
From playlist Discrete Math II/Combinatorics (entire course)
Graph Theory: 64. Vertex Colouring
In this video we define a (proper) vertex colouring of a graph and the chromatic number of a graph. We discuss some basic facts about the chromatic number as well as how a k-colouring partitions the vertex set into k independent sets (check out video #50 for more about independent sets).
From playlist Graph Theory part-11
Edge Coloring and the Chromatic Index of a Graph
This video introduces edge coloring and the chromatic index of a graph. An application of the chromatic index is provided. mathispower4u.com
From playlist Graph Theory (Discrete Math)
Math for Liberal Studies - Lecture 1.7.3 Upper and Lower Bounds
This is the third and final video lecture for Math for Liberal Studies Section 1.7: Coloring Graphs. In this video, we discuss the "chromatic number" for a graph, which is the smallest number of colors needed to properly color the vertices of the graph. In general, finding the exact chroma
From playlist Math for Liberal Studies Lectures
Math for Liberal Studies: The Greedy Coloring Algorithm
In this video, we use the Greedy Coloring Algorithm to solve a couple of graph coloring problems. For more info, visit the Math for Liberal Studies homepage: http://webspace.ship.edu/jehamb/mls/index.html
From playlist Math for Liberal Studies
What are Connected Graphs? | Graph Theory
What is a connected graph in graph theory? That is the subject of today's math lesson! A connected graph is a graph in which every pair of vertices is connected, which means there exists a path in the graph with those vertices as endpoints. We can think of it this way: if, by traveling acr
From playlist Graph Theory
This is Lecture 23 of the CSE547 (Discrete Mathematics) taught by Professor Steven Skiena [http://www.cs.sunysb.edu/~skiena/] at Stony Brook University in 1999. The lecture slides are available at: http://www.cs.sunysb.edu/~algorith/math-video/slides/Lecture%2023.pdf More information may
From playlist CSE547 - Discrete Mathematics - 1999 SBU
Chromatic Number of Complete Graphs | Graph Theory
What are the chromatic numbers of complete graphs on n vertices? As we’ll see in today’s graph theory lesson on vertex coloring, we need exactly n colors to properly color the complete graph K_n. Intro to Graph Colorings: https://youtu.be/3VeQhNF5-rE Recall that a proper coloring (or ju
From playlist Graph Theory
How to Tell if Graph is Bipartite (by hand) | Graph Theory
How can we tell if a graph is bipartite by hand? We'll discuss the easiest way to identify bipartite graphs in today's graph theory lesson. This method takes advantage of the fact that bipartite graphs are 2-colorable. This means their vertices can be colored using only two colors so adjac
From playlist Graph Theory
Louis Esperet: Coloring graphs on surfaces
Recording during the thematic meeting: "Graphs and surfaces: algorithms, combinatorics and topology" the May 11, 2016 at the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematici
From playlist Mathematical Aspects of Computer Science
On the effect of randomness on planted 3-coloring models - Uri Feige
Computer Science/Discrete Mathematics Seminar I Topic: On the effect of randomness on planted 3-coloring models Speaker: Uri Feige Affiliation: Weizmann Institute of Science Date: Monday, November 21 For more video, visit http://video.ias.edu
From playlist Mathematics
Puzzle 10: A Weekend To Remember
MIT 6.S095 Programming for the Puzzled, IAP 2018 View the complete course: https://ocw.mit.edu/6-S095IAP18 Instructor: Srini Devadas You are happy when your friends are happy. This means making sure that some pairs of your friends never meet at any of your parties. This video will explain
From playlist MIT 6.S095 Programming for the Puzzled, January IAP 2018
Maria Chudnovsky: Coloring graphs with forbidden induced paths
Abstract: The problem of testing if a graph can be colored with a given number k of colors is NP-complete for every k[greater than]2. But what if we have more information about the input graph, namely that some fixed graph H is not present in it as an induced subgraph? It is known that the
From playlist Combinatorics
Chromatic Number of Bipartite Graphs | Graph Theory
What is the chromatic number of bipartite graphs? If you remember the definition, you may immediately think the answer is 2! This is practically correct, though there is one other case we have to consider where the chromatic number is 1. We'll explain both possibilities in today's graph th
From playlist Graph Theory
The Colorful Connected Subgraph Problem - Richard Karp
A Celebration of Mathematics and Computer Science Celebrating Avi Wigderson's 60th Birthday October 5 - 8, 2016 More videos on http://video.ias.edu
From playlist Mathematics
Graph Coloring is NP-Complete - Intro to Algorithms
This video is part of an online course, Intro to Algorithms. Check out the course here: https://www.udacity.com/course/cs215.
From playlist Introduction to Algorithms