# How to Find Chromatic Number | Graph Coloring Algorithm

## Chromatic Number-

Before you go through this article, make sure that you have gone through the previous article on Chromatic Number.

We gave discussed-

• Graph Coloring is a process of assigning colors to the vertices of a graph.
• It ensures that no two adjacent vertices of the graph are colored with the same color.
• Chromatic Number is the minimum number of colors required to properly color any graph.

In this article, we will discuss how to find Chromatic Number of any graph.

## Graph Coloring Algorithm-

• There exists no efficient algorithm for coloring a graph with minimum number of colors.
• Graph Coloring is a NP complete problem.

However, a following greedy algorithm is known for finding the chromatic number of any given graph.

## Greedy Algorithm-

### Step-01:

Color first vertex with the first color.

### Step-02:

Now, consider the remaining (V-1) vertices one by one and do the following-

• Color the currently picked vertex with the lowest numbered color if it has not been used to color any of its adjacent vertices.
• If it has been used, then choose the next least numbered color.
• If all the previously used colors have been used, then assign a new color to the currently picked vertex.

## Drawbacks of Greedy Algorithm-

There are following drawbacks of the above Greedy Algorithm-

• The above algorithm does not always use minimum number of colors.
• The number of colors used sometimes depend on the order in which the vertices are processed.

Also Read- Types of Graphs in Graph Theory

## Problem-01:

Find chromatic number of the following graph- ## Solution-

Applying Greedy Algorithm, we have-

 Vertex a b c d e f Color C1 C2 C1 C2 C1 C2

From here,

• Minimum number of colors used to color the given graph are 2.
• Therefore, Chromatic Number of the given graph = 2.

The given graph may be properly colored using 2 colors as shown below- ## Problem-02:

Find chromatic number of the following graph- ## Solution-

Applying Greedy Algorithm, we have-

 Vertex a b c d e f Color C1 C2 C2 C3 C3 C1

From here,

• Minimum number of colors used to color the given graph are 3.
• Therefore, Chromatic Number of the given graph = 3.

The given graph may be properly colored using 3 colors as shown below- ## Problem-03:

Find chromatic number of the following graph- ## Solution-

Applying Greedy Algorithm, we have-

 Vertex a b c d e f g Color C1 C2 C1 C3 C2 C3 C4

From here,

• Minimum number of colors used to color the given graph are 4.
• Therefore, Chromatic Number of the given graph = 4.

The given graph may be properly colored using 4 colors as shown below- ## Problem-04:

Find chromatic number of the following graph- ## Solution-

Applying Greedy Algorithm, we have-

 Vertex a b c d e f Color C1 C2 C3 C1 C2 C3

From here,

• Minimum number of colors used to color the given graph are 3.
• Therefore, Chromatic Number of the given graph = 3.

The given graph may be properly colored using 3 colors as shown below- ## Problem-05:

Find chromatic number of the following graph- ## Solution-

Applying Greedy Algorithm,

• Minimum number of colors required to color the given graph are 3.
• Therefore, Chromatic Number of the given graph = 3.

The given graph may be properly colored using 3 colors as shown below- To gain better understanding about How to Find Chromatic Number,

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Summary Article Name
How to Find Chromatic Number | Graph Coloring Algorithm
Description
Graph Coloring Algorithm- A Greedy Algorithm exists for Graph Coloring.How to find Chromatic Number of a graph- We follow the Greedy Algorithm to find Chromatic Number of the Graph. Problems on finding Chromatic Number of a given graph.
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Gate Vidyalay
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