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Q3E

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Discrete Mathematics and its Applications
Found in: Page 733
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

Construct the dual graph for the map shown:

Then find the number of colors needed to color the map so that no two adjacent regions have the same color.

The dual graph for the map given is:

And the number of colors needed to color the given map so that no two adjacent regions have the same color is \(3\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given data in the question.

In the question, a map is given.

Step 2: Definition to be used.

According to the definition of dual graph, each map can be represented by a graph and in every graph each region of the map is represented by a vertex also two vertices are connected by a edge if and only if the region represented by these vertices have a common border, then the obtained graph is known as dual graph.

Step 3: Construct the dual graph for given map.

Construct the dual graph of the given map by determining one vertex in each region as follows:

Step 4: Find the number of colors needed to color the map.

Since, the vertex \(a\) is connected to all the other vertices, thus the color of the vertex \(a\) should be unique or different from all the other vertices and the vertices \(b\) and \(d\) are not directly connected to each other similarly vertices \(f,c\) and \(e\) are not directly connected to each other.

Therefore,

Region

Vertex

Color

A

a

Blue

B

b

Green

C

c

Red

D

d

Green

E

e

Red

F

f

Red

Color the given map as shown:

Hence, \[3\] color are needed to color the given map.

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