A graph is a collection (nonempty set) of vertices and edges. A graph G is a set of vertex (nodes) v connected by edges (links) e. Thus G=(v , e).

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Theoretical Section
Practical Section
INTRODUCTION TO GRAPH AND GRAPH
COLORING PROBLEM
Design Methods and Analysis of Algorithm
Darwish Ahmad Herati
St. Joseph’s College (Autonomous)
Computer Science Department
MSc (Computer Science)
SUPERVISOR
Prof. Ms. Mrinmoyee Bhattacharya
September 1, 2015
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Contents
1 Theoretical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
2 Practical Section
Examples
Implementation
Simulation Technologies
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Introduction to Graph:
A graph is a collection (nonempty set) of vertices and
edges
A graph G is a set of vertex (nodes) v connected by edges
(links) e. Thus G=(v , e).
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Example of Graphs:
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Introduction to Graphs:
A graph is a mathematical object that is used to model
different situations objects and processes:
- Linked list
- Tree(Special type of graph)
- Flowchart chart of a program
- Structure chart of a program
- Finite state automata
- Electric Circuits
- Course Curriculum
- etc.
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Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Vertex (Node):
Edge (Link):
Adjacent Vertices:
Sub-Graph:
Directed Graph:
Undirected Graph:
Connected Graph:
Unconnected Graph:
Paths:
Simple path:
Cycles:
Loop:
Trees:
Spanning tree of a graph:
Complete graphs:
Weighted graphs:
Networks:
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Chromatic Number:
K-coloring:
Optimal Coloring:
etc.
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Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Vertex (Node): can have names and properties
Edge (Link): connect two vertices, can be labeled, can be
directed
Adjacent Vertices: if there is an edge between them.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Subgraph: A subgraph S of a graph G is a graph whose
set of vertices and set of edges are all subsets of G.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Directed Graph: In directed
graphs the edges are oriented,
they have a beginning and an
end. Thus A B and B A are
different edges.
Sometimes the edges of a
directed graph are called arcs.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Undirected Graph: In
undirected graphs the edges are
symmetrical, e.g. if A and B are
vertices, A B and B A are one
and the same edge. Graph1
above is undirected.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Connected Graph: There is a path between each two
vertices.
Unconnected Graph: There are at least two vertices not
connected by a path.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
paths: A path is a list of vertices in which successive
vertices are connected by edges
Simple Path: No vertex is repeated.
Some paths in Graph1 :
A B C D
A C B A C D
A B
D C B
C B A
Some paths in Graph2:
D A B
A D A C
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Cycles:A cycle is a simple path with distinct edges, where
the first vertex is equal to the last.
Cycles in Graph1:
C A B C, C B A C, A B C A, A C B A, B A C B, B C A B A B
A is not a cycle, because the edge A B is the same as B A
Loop: An edge that connects the vertex with itself
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Trees:A tree is an undirected graph with no cycles and a
vertex chosen to be the root of the tree.
Note: in a tree, when we choose a root we impose an
orientation. Given an acyclic graph, we may choose any
node to be the root of a tree.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Spanning Tree of a Graph: A spanning tree of an
undirected graph is a subgraph that contains all the
vertices, and no cycles. If we add any edge to the spanning
tree, it forms a cycle, and the tree becomes a graph.
It is possible to define a spanning tree for directed graphs,
however the definition is rather complicated and will not be
discussed here.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Complete Graphs:Graphs with all edges present each
vertex is connected to all other vertices, are called
complete graphs.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Weighted Graphs: Weights are
assigned to each edge
(e.g. distances in a road map)
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Networks:Directed weighted
graphs
Note:Some textbooks define
networks to be undirected
weighted graphs as well.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Terminologies of Graphs:
Chromatic Number: The chromatic number of a graph G
is the smallest number k for which a k-coloring of the
vertices of G is possible.We will use the notation x(G) to
denote the chromatic number of G.
k-coloring: A k-coloring of a graph G is a coloring of the
vertices of G using k colors and satisfying the requirement
that adjacent vertices are colored with different colors.
Optimal Coloring: An optimal coloring of a graph G is a
coloring of the vertices of G using the fewest possible
number of colors.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Introduction to Graph Coloring
Graph coloring is one of the oldest concepts in the theory
of graphs,
A graph G = (V, E) consists two sets where one is the set
of vertices and another is the set of edges such that each
edges is associated with an un ordered pair of vertices and
graph coloring is one of the most useful models in graph
theory. Graph coloring is the way of coloring the vertices of
a graph with the minimum number of colors such that no
two adjacent vertices share the same color.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Introduction to Graph Coloring
A k-coloring of graph G is an assignment of integers
{1, 2, . . . ,k} (the colors) to the vertices of G in such a way
that neighbors receive different integers. The chromatic
number of G is the smallest k such that G has a k-coloring.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Types of Graph Coloring:
Vertix Coloring:
Edge Coloring:
Face Coloring/ Map Coloring:
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Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Vertix Coloring of Graph:
Vertix Coloring:A vertex coloring of a graph is to color the
vertices of the graph in such a way that any two adjacent
vertices receive different colors.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Edge Coloring of Graph:
Edge Coloring:An edge coloring of a graph is to color the
edges of the graph in such a way that any two adjacent
edges receive different colors.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Face Coloring / Map Coloring of Graph:
Face Coloring/ Map Coloring:
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Francis Guthrie and Demorgan
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Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Cayley and Kempe 1879
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Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Percy John Heawood 1890
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Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
K. Appel W. Haken 1977
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Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Rebertson, Sanders, Seymour, Thomas
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Definitions
Algorithm
Complexity
Applications
Definitions:
An n-coloring is proper if no pair of adjacent vertices gets
the same color.
The graph coloring problem involves assigning values (or
colors) to the vertices of a graph so that adjacent vertices
are assigned distinct colors. With the objective of
minimizing the number of colors used.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Greedy Algorithm for Graph Coloring:
Step 1. Assign the first color (c1) to the first vertex (v1).
Step 2. Vertex v2 is assigned color c1 if it is not adjacent
to v1; otherwise it gets assigned color c2.
Steps 3,4,...,n. Vertex v1 is assigned the first possible
color in the priority list of colors (i.e. the first color that has
not been assigned to one of the already colored neighbors
of vi ).
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Analysis of Graph Coloring:
An Upper bound on the computing time of MCOLORING
can arrived by finding the number of internal nodes in the
state space tree is:
At each internal node, O(mn) time is spent by Next Value
to determine the children corresponding to legal coloring.
Hence the total time is bounded by:
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Applications of Graph Coloring:
Graph Coloring: has many applications in job
scheduling, assignments of classes/classrooms,
assignments of wireless channels.
Aircraft scheduling:
Making Schedule or Time Table:
Mobile Radio Frequency Assignment:
Suduku:
Register Allocation:
Bipartite Graphs:
Map Coloring:
etc.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Aircraft Scheduling:
Aircraft scheduling: Assume that we have k aircrafts,
and we have to assign them to n flights, where the ith flight
is during the time interval (ai, bi). Clearly, if two flights
overlap, then we cannot assign the same aircraft to both
flights. The vertices of the conflict graph correspond to the
flights, two vertices are connected if the corresponding
time intervals overlap. Therefore the conflict graph is an
interval graph, which can be colored optimally in
polynomial time.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Making Schedule or Time Table:
Making Schedule or Time Table: Suppose we want to
make am exam schedule for a university. We have list
different subjects and students enrolled in every subject.
Many subjects would have common students (of same
batch, some backlog students, etc). How do we schedule
the exam so that no two exams with a common student are
scheduled at same time? How many minimum time slots
are needed to schedule all exams? This problem can be
represented as a graph where every vertex is a subject
and an edge between two vertices mean there is a
common student. So this is a graph coloring problem
where minimum number of time slots is equal to the
chromatic number of the graph.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Mobile Radio Frequency Assignment:
Mobile Radio Frequency Assignment: When
frequencies are assigned to towers, frequencies assigned
to all towers at the same location must be different. How to
assign frequencies with this constraint? What is the
minimum number of frequencies needed? This problem is
also an instance of graph coloring problem where every
tower represents a vertex and an edge between two towers
represents that they are in range of each other.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Suduku:
Suduku: Suduku is also a variation of Graph coloring
problem where every cell represents a vertex. There is an
edge between two vertices if they are in same row or same
column or same block.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Register Allocation:
Register Allocation: In compiler optimization, register
allocation is the process of assigning a large number of
target program variables onto a small number of CPU
registers. This problem is also a graph coloring problem.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Bipartite:
Bipartite Graphs: We can check if a graph is Bipartite or
not by colowing the graph using two colors. If a given
graph is 2-colorable, then it is Bipartite, otherwise not. See
this for more details.
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Theoretical Section
Practical Section
Introduction
History
Definitions
Algorithm
Complexity
Applications
Map Coloring:
Map Coloring: Geographical maps of countries or states
where no two adjacent cities cannot be assigned same
color. Four colors are sufficient to color any map (See Four
Color Theorem)
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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Coloring Examples:
: Vertex Coloring
: Vertix Coloring
: Vertix Coloring
: Edge Coloring
: Both Vertex & Edge
: Face Coloring
: Map Coloring
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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Coloring Examples:
: Vertex Coloring
: Vertix Coloring
: Vertix Coloring
: Edge Coloring
: Both Vertex & Edge
: Face Coloring
: Map Coloring
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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Coloring Examples:
: Vertex Coloring
: Vertix Coloring
: Vertix Coloring
: Edge Coloring
: Both Vertex & Edge
: Face Coloring
: Map Coloring
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Coloring Examples:
: Vertex Coloring
: Vertix Coloring
: Vertix Coloring
: Edge Coloring
: Both Vertex & Edge
: Face Coloring
: Map Coloring
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Coloring Examples:
: Vertex Coloring
: Vertix Coloring
: Vertix Coloring
: Edge Coloring
: Both Vertex & Edge
: Face Coloring
: Map Coloring
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Coloring Examples:
: Vertex Coloring
: Vertix Coloring
: Vertix Coloring
: Edge Coloring
: Both Vertex & Edge
: Face Coloring
: Map Coloring
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Coloring Examples:
: Vertex Coloring
: Vertix Coloring
: Vertix Coloring
: Edge Coloring
: Both Vertex & Edge
: Face Coloring
: Map Coloring
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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Implementation of Graphs using C++:
Implementation:Two Graphs using Greedy Method
Implementation in C++.
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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Theorem Simulation Technologies:
1st: Demonstration
Program
2nd: CGraph Simulator
3rd: GraphTea
Simulator
4th: Graph Magic
5th: etc.
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Theorem Simulation Technologies:
1st: Demonstration
Program
2nd: CGraph Simulator
3rd: GraphTea
Simulator
4th: Graph Magic
5th: etc.
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Theorem Simulation Technologies:
1st: Demonstration
Program
2nd: CGraph Simulator
3rd: GraphTea
Simulator
4th: Graph Magic
5th: etc.
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Theorem Simulation Technologies:
1st: Demonstration
Program
2nd: CGraph Simulator
3rd: GraphTea
Simulator
4th: Graph Magic
5th: etc.
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
Graph Theorem Simulation Technologies:
1st: Demonstration
Program
2nd: CGraph Simulator
3rd: GraphTea
Simulator
4th: Graph Magic
5th: etc.
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Theoretical Section
Practical Section
Examples
Implementation
Simulation Technologies
References
Srikanth.S, (2014). Design and Analysis of Algorithms
Tommy R. Jensen and Bjarne Toft, (1995) Graph Coloring
Problems
Lydia Sinapova. (2015)
(http://faculty.simpson.edu/lydia.sinapova/www/cmsc250/LN250_W
Graphs.htm) . Accessed Augest 25
2015.
etc.
Darwish Ahmad Herati Design Methods and analysis of Algorithm

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Practical Section
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Simulation Technologies
Thank You For Your
Attention
Darwish Ahmad Herati Design Methods and analysis of Algorithm