Question

Find the chromatic number of the given graph.

Short answer

The Chromatic number of the given graph is \(3\).

Step-by-step solution

Given data in the question.

In the question, a graph is given.

Definition to be used.

According to the definition of chromatic number of a graph, the chromatic number of a graph is the least number of colors needed to color a graph so that no two adjacent vertices are assigned the same color.

Find the chromatic number of the given graph.

Since, the vertex \(g\) is connected to all other vertices so, the color of vertex \(g\) should be unique or different from all other vertices and the no pair of vertices \(b,d\) and \(f\) are connected to each other directly, thus the color of vertices \(b,d\) and \(f\) can be same similarly, no pairs of vertices \(a,c\) and \(e\) are directly connected to each other so the color of vertices \(a,c\) and \(e\) can be same.

Therefore,

Vertex	Color
a	Green
b	Red
c	Green
d	Red
e	Green
f	Red
g	Blue

Hence, \(3\) different colors are used to color the given graph. So, the chromatic number of the given graph is \(3\).