Hello all, This is the presentation of Graph Colouring in Graph theory and application. Use this presentation as a reference if you have any doubt you can comment here.
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 share 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 share 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.
Hello all, This is the presentation of Graph Colouring in Graph theory and application. Use this presentation as a reference if you have any doubt you can comment here.
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 share 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 share 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.
Graph theory - Traveling Salesman and Chinese PostmanChristian Kehl
Traveling Salesman and Chinese Postman problems
1. Problem Description and Complexity
2. Theoretical Approach
3. Practical Approaches and Possible Solutions
4. Examples
NP Complete problems in the field of graph theory have been selected and have been tested for a polynomial solution. Successfully studied and implemented a few solutions to various NP-Complete Problems. Various polynomial time reductions are also been studied between these problems and and methods have been worked on. I have secured a letter of appreciation from the Guide for my performance during the course of the Internship.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
Discussed Elements of Dynamic Programming, covered all the points from Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein,‖
Introduction to Algorithms‖, Third Edition, Prentice-Hall, 2011.
Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can
divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The
condition which we follow in graph coloring is that the incident vertices/edges have not the same color.
There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and
others are online algorithm. Where offline means the graph is known in advance and the online means that
the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is
added to the graph and the main issue is that we want to minimize the number of colors. We cannot change
the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm
for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is Δ, then it
is possible to color its edges, in polynomial time, using at most Δ+ 1 color. The algorithm provided by
Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one
by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of
Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors
online in multiple rounds.
Chromatic Number of a Graph (Graph Colouring)Adwait Hegde
A graph coloring is an assignment of labels, called colors, to the vertices of a graph such that no two adjacent vertices share the same color. The chromatic number χ(G) of a graph G is the minimal number of colors for which such an assignment is possible.
Graph theory - Traveling Salesman and Chinese PostmanChristian Kehl
Traveling Salesman and Chinese Postman problems
1. Problem Description and Complexity
2. Theoretical Approach
3. Practical Approaches and Possible Solutions
4. Examples
NP Complete problems in the field of graph theory have been selected and have been tested for a polynomial solution. Successfully studied and implemented a few solutions to various NP-Complete Problems. Various polynomial time reductions are also been studied between these problems and and methods have been worked on. I have secured a letter of appreciation from the Guide for my performance during the course of the Internship.
Divide and Conquer Algorithms - D&C forms a distinct algorithm design technique in computer science, wherein a problem is solved by repeatedly invoking the algorithm on smaller occurrences of the same problem. Binary search, merge sort, Euclid's algorithm can all be formulated as examples of divide and conquer algorithms. Strassen's algorithm and Nearest Neighbor algorithm are two other examples.
Discussed Elements of Dynamic Programming, covered all the points from Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein,‖
Introduction to Algorithms‖, Third Edition, Prentice-Hall, 2011.
Graph coloring is the assignment of colors to the graph vertices and edges in the graph theory. We can
divide the graph coloring in two types. The first is vertex coloring and the second is edge coloring. The
condition which we follow in graph coloring is that the incident vertices/edges have not the same color.
There are some algorithms which solve the problem of graph coloring. Some are offline algorithm and
others are online algorithm. Where offline means the graph is known in advance and the online means that
the edges of the graph are arrive one by one as an input, and We need to color each edge as soon as it is
added to the graph and the main issue is that we want to minimize the number of colors. We cannot change
the color of an edge after colored in an online algorithm. In this paper, we improve the online algorithm
for edge coloring. There is also a theorem which proves that if the maximum degree of a graph is Δ, then it
is possible to color its edges, in polynomial time, using at most Δ+ 1 color. The algorithm provided by
Vizing is offline, i.e., it assumes the whole graph is known in advance. In online algorithm edges arrive one
by one in a random permutation. This online algorithm is inspired by a distributed offline algorithm of
Panconesi and Srinivasan, referred as PS algorithm, works on 2-rounds which we extend by reusing colors
online in multiple rounds.
Chromatic Number of a Graph (Graph Colouring)Adwait Hegde
A graph coloring is an assignment of labels, called colors, to the vertices of a graph such that no two adjacent vertices share the same color. The chromatic number χ(G) of a graph G is the minimal number of colors for which such an assignment is possible.
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.
Greedy Edge Colouring for Lower Bound of an Achromatic Index of Simple Graphsinventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Graph coloring is an important concept in graph theory. It is a special kind of problem in which we have assign colors to certain elements of the graph along with certain constraints. Suppose we are given K colors, we have to color the vertices in such a way that no two adjacent vertices of the graph have the same color, this is known as vertex coloring, similarly we have edge coloring and face coloring. The coloring problem has a huge number of applications in modern computer science such as making schedule of time table , Sudoku, Bipartite graphs , Map coloring, data mining, networking. In this paper we are going to focus on certain applications like Final exam timetabling, Aircraft Scheduling, guarding an art gallery.
Analysis and design of algorithms part 4Deepak John
Complexity Theory - Introduction. P and NP. NP-Complete problems. Approximation algorithms. Bin packing, Graph coloring. Traveling salesperson Problem.
Parallel Algorithm for Graph Coloring Heman Pathak
The graph coloring problem is an assignment of colors to the vertices such that no two adjacent vertices are assigned the same color. A k-coloring of a graph G is a coloring of G using k colors.
SATISFIABILITY METHODS FOR COLOURING GRAPHScscpconf
The graph colouring problem can be solved using methods based on Satisfiability (SAT). An instance of the problem is defined by a Boolean expression written using Boolean variables and the logical connectives AND, OR and NOT. It has to be determined whether there is an assignment of TRUE and FALSE values to the variables that makes the entire expression true.A SAT problem is syntactically and semantically quite simple. Many Constraint Satisfaction Problems (CSPs)in AI and OR can be formulated in SAT. These make use of two kinds of
searchalgorithms: Deterministic and Randomized.It has been found that deterministic methods when run on hard CSP instances are frequently very slow in execution.A deterministic method always outputs a solution in the end, but it can take an enormous amount of time to do so.This has led to the development of randomized search algorithms like GSAT, which are typically based on local (i.e., neighbourhood) search. Such methodshave been applied very successfully to find good solutions to hard decision problems
Presentation at evostar 2015.
Abstract: Mathematical morphology (MM) is broadly used in image processing. MM operators require to establish an order between the values of a set of pixels. This is why MM is basically used with binary and grayscale images. Many works have been focused on extending MM to colour images by mapping a multi-dimensional colour space onto a linear ordered space. However, most of them are not validated in terms of topology preservation but in terms of the results once MM operations are applied. This work presents an evolutionary method to obtain total- and P-orderings of a colour space, i.e. RGB, maximising topology preservation. This approach can be used to order a whole colour space as well as to get a specific ordering for the subset of colours appearing in a particular image. These alternatives improve the results obtained with the orderings usually employed, in both topology preservation and noise reduction.
Analysis & Design of Algorithms
Backtracking
N-Queens Problem
Hamiltonian circuit
Graph coloring
A presentation on unit Backtracking from the ADA subject of Engineering.
undirected graph and a number m, determine if the graph can be colored with at most m colors such that no two adjacent vertices of the graph are colored with the same color.
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Similar to Graph Coloring : Greedy Algorithm & Welsh Powell Algorithm (20)
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
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• Compatible with MAFI CCR system
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• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
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• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
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1. Maulana Azad NationalMaulana Azad National InstituteInstitute ofof
TechnologyTechnology
Department of Computer Science & EngineeringDepartment of Computer Science & Engineering
PresentationPresentation
OnOn
Graph ColoringGraph Coloring
Presented By:Presented By:
Priyank JainPriyank Jain
Shweta SaxenaShweta Saxena
2. What is Graph Coloring?What is Graph Coloring?
Graph Coloring is an assignment of colorsGraph Coloring is an assignment of colors
(or any distinct marks) to the vertices of a(or any distinct marks) to the vertices of a
graph. Strictly speaking, a coloring is agraph. Strictly speaking, a coloring is a
proper coloring if no two adjacent verticesproper coloring if no two adjacent vertices
have the same color.have the same color.
5. Why Graph Coloring?Why Graph Coloring?
Many problems can be formulated as aMany problems can be formulated as a
graph coloring problem including Timegraph coloring problem including Time
Tabling,Tabling, Channel AssignmentChannel Assignment etc.etc.
A lot of research has been done in thisA lot of research has been done in this
area.area.
6. Channel AssignmentChannel Assignment
Find a channel assignment to R radioFind a channel assignment to R radio
stations such that no station has a conflictstations such that no station has a conflict
(there is a conflict if they are in vicinity)(there is a conflict if they are in vicinity)
Vertices – radio stations, edges – conflict,Vertices – radio stations, edges – conflict,
colors – available channelscolors – available channels
7. TerminologyTerminology
K-ColoringK-Coloring
A k-coloring of a graph G is a mapping ofA k-coloring of a graph G is a mapping of
V(G) onto the integers 1..k such that adjacentV(G) onto the integers 1..k such that adjacent
vertices map into different integers.vertices map into different integers.
A k-coloring partitions V(G) into k disjointA k-coloring partitions V(G) into k disjoint
subsets such that vertices from differentsubsets such that vertices from different
subsets have different colors.subsets have different colors.
8. TerminologyTerminology
K-colorableK-colorable
A graph G is k-colorable if it has a k-coloring.A graph G is k-colorable if it has a k-coloring.
Chromatic NumberChromatic Number
The smallest integer k for which G is k-The smallest integer k for which G is k-
colorable is called the chromatic number of G.colorable is called the chromatic number of G.
9. TerminologyTerminology
K-chromatic graphK-chromatic graph
A graph whose chromatic number is k isA graph whose chromatic number is k is
called a k-chromatic graph.called a k-chromatic graph.
ColoringColoring
A coloring of a graph G assigns colors to theA coloring of a graph G assigns colors to the
vertices of G so that adjacent vertices arevertices of G so that adjacent vertices are
given different colorsgiven different colors
10. Types of Graph ColoringTypes of Graph Coloring
Vertex ColoringVertex Coloring
Edge ColoringEdge Coloring
12. ExampleExample
Problem: A state legislature has aProblem: A state legislature has a
number of committees that meet eachnumber of committees that meet each
week for one hour. How can we scheduleweek for one hour. How can we schedule
the committee meetings times such thatthe committee meetings times such that
the least amount of time is used but suchthe least amount of time is used but such
that two committees with overlappingthat two committees with overlapping
membership do not meet at the samemembership do not meet at the same
time.time.
13. Example (cont)Example (cont)
The chromatic number of this graph is four. Thus four hours suffice to schedule
committee meetings without conflict.
An edge represents a conflict between to meetings
An vertex represents a meeting
14. Graph Colouring AlgorithmGraph Colouring Algorithm
There is no efficient algorithm available forThere is no efficient algorithm available for
coloring a graph with minimum number ofcoloring a graph with minimum number of
colors.colors.
Graph coloring problem is a known NPGraph coloring problem is a known NP
Complete problem.Complete problem.
15. NP Complete ProblemNP Complete Problem
NP complete problems are problemsNP complete problems are problems
whose status is unknown.whose status is unknown.
No polynomial time algorithm has yetNo polynomial time algorithm has yet
been discovered for any NP completebeen discovered for any NP complete
problemproblem
It is not established that no polynomial-It is not established that no polynomial-
time algorithm exist for any of them.time algorithm exist for any of them.
16. NP Complete ProblemNP Complete Problem
The interesting part is, if any one of theThe interesting part is, if any one of the
NP complete problems can be solved inNP complete problems can be solved in
polynomial time, then all of them can bepolynomial time, then all of them can be
solved.solved.
Although Graph coloring problem is NPAlthough Graph coloring problem is NP
Complete problem there are someComplete problem there are some
approximate algorithms to solve the graphapproximate algorithms to solve the graph
coloring problem.coloring problem.
17. Basic Greedy AlgorithmBasic Greedy Algorithm
1.1. Color first vertex with first color.Color first vertex with first color.
2. Do following for remaining V-1 vertices.2. Do following for remaining V-1 vertices.
a)a) Consider the currently picked vertexConsider the currently picked vertex
and color it with the lowest numberedand color it with the lowest numbered
color that has not been used on anycolor that has not been used on any
previously colored vertices adjacent to it.previously colored vertices adjacent to it.
If all previously used colors appear onIf all previously used colors appear on
vertices adjacent to v, assign a new colorvertices adjacent to v, assign a new color
to it.to it.
18. Analysis of Greedy AlgorithmAnalysis of Greedy Algorithm
The above algorithm doesn’t always useThe above algorithm doesn’t always use
minimum number of colors. Also, theminimum number of colors. Also, the
number of colors used sometime dependnumber of colors used sometime depend
on the order in which vertices areon the order in which vertices are
processedprocessed
19. Example:Example:
For example, consider the following twoFor example, consider the following two
graphs. Note that in graph on right side,graphs. Note that in graph on right side,
vertices 3 and 4 are swapped. If wevertices 3 and 4 are swapped. If we
consider the vertices 0, 1, 2, 3, 4 in leftconsider the vertices 0, 1, 2, 3, 4 in left
graph, we can color the graph using 3graph, we can color the graph using 3
colors. But if we consider the vertices 0, 1,colors. But if we consider the vertices 0, 1,
2, 3, 4 in right graph, we need 4 colors2, 3, 4 in right graph, we need 4 colors
21. WelshWelsh PowellPowell AlgorithmAlgorithm
Find the degree of each vertexFind the degree of each vertex
ListList the vericesthe verices in order of descendingin order of descending
valence i.e.valence i.e. degree(v(i))>=degree(v(i+1))degree(v(i))>=degree(v(i+1))
ColourColour the first vertex in the listthe first vertex in the list
Go down the sorted list and color everyGo down the sorted list and color every
vertex not connected to the coloredvertex not connected to the colored
vertices above the same color then crossvertices above the same color then cross
out all colored vertices in the list.out all colored vertices in the list.
22. Welsh Powell AlgorithmWelsh Powell Algorithm
Repeat the process on the uncoloredRepeat the process on the uncolored
vertices with a new color-always workingvertices with a new color-always working
in descending order of degree until allin descending order of degree until all
vertices are colored.vertices are colored.
ComplexityComplexity of above algorithm =of above algorithm = O(nO(n22
))
Note: Each color defines an independent set of vertices ( vertices with no edges between them. ).
Note: In order to verify that the chromatic number of a graph is a number k, we must also show that the graph can not be properly colored with k-1 colors. In other words the goal is to show that the (k-1)-coloring we might construct for the graph must force two adjacent vertices to have the same color.
Look at previous example. Simply state that the since we found the chromatic number to be N the graph is N-chromatic.
For k-colorable: look at example and say that since it has a 6-coloring therefore it is
6-colorable.
For chromatic number: Look at previous example. See if students can find a better coloring of G and state its chromatic number. This is a k-chromatic graph!