**Task:**2. For each of the following graphs, find the chromatic index and provide a coloring that demonstrates it is possible. In your drawing, make sure the graphs are large enough so that it is clear what the colors are!**Diagrams:**(a) **Graph Description:**- This graph consists of an outer pentagon with five vertices. - Each vertex of the pentagon is connected to every other vertex, forming a complete graph (K5).- Inside the pentagon is another smaller pentagon with its own set of five vertices.- The vertices of the inner pentagon are connected to the vertices of the outer pentagon by the corresponding edges.- The overall structure resembles a pentagonal star or a two-layered graph with a total of 10 vertices interconnected.(b) **Graph Description:**- This graph also has an outer pentagon with five vertices.- Inside the outer pentagon is a pentagram (a five-pointed star) with its vertices aligned with the pentagon vertices.- The pentagram's points are connected internally, forming a smaller inner pentagon.- The structure is intertwined, typically resulting in a colorful pattern due to the crossing lines.(c) **Graph Description:**- The graph is a star with eight rays.- There is one central vertex connected to eight outer vertices.- Each outer vertex is connected only to the central vertex, resembling a sun-like structure.

Answered: 2. For each of the following graphs,…

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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**Task:**

2. For each of the following graphs, find the chromatic index and provide a coloring that demonstrates it is possible. In your drawing, make sure the graphs are large enough so that it is clear what the colors are!

**Diagrams:**

(a) **Graph Description:**
- This graph consists of an outer pentagon with five vertices.
- Each vertex of the pentagon is connected to every other vertex, forming a complete graph (K5).
- Inside the pentagon is another smaller pentagon with its own set of five vertices.
- The vertices of the inner pentagon are connected to the vertices of the outer pentagon by the corresponding edges.
- The overall structure resembles a pentagonal star or a two-layered graph with a total of 10 vertices interconnected.

(b) **Graph Description:**
- This graph also has an outer pentagon with five vertices.
- Inside the outer pentagon is a pentagram (a five-pointed star) with its vertices aligned with the pentagon vertices.
- The pentagram's points are connected internally, forming a smaller inner pentagon.
- The structure is intertwined, typically resulting in a colorful pattern due to the crossing lines.

(c) **Graph Description:**
- The graph is a star with eight rays.
- There is one central vertex connected to eight outer vertices.
- Each outer vertex is connected only to the central vertex, resembling a sun-like structure.

Expert Solution

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Solution-

Chromatic number is the bare minimum of colours needed to colour a graph so that no two of its neighbouring vertices have the same colour allocated to them.

The chromatic number of any given graph can be found using the greedy technique described below.

let colors are represented by 0,1,2,3,4,5,14 and each vertex is given the color with the smallest member that is not

already used by one of its neighbors.

Greedy Algorithm:

Step 1: Color first nexter with the fisst color. step: consider the remaining vertex One by one and do the following

i) Color the currently picked vertex with the lowest numbered color if it has not been used to color any of its adjacent vertices.

ii) if it has been used, then choose the next least numbered color .

iii) if all the previously used colors have been used then assign a new Color to the currently picked vertex.

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