4 colors required

1 |
2 |
3 |
4 |

A | G | F | D |

H | C | B | |

E |

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Now to find the chromatic no of the graph we need to find the chromatic partitioning of this graph.

So, to find chromatic partitioning of a graph G enumerate through all maximal independent set of graph G and take the minimum number of these sets which collectively include all vertices of G.

The maximal independent sets of this graph are

{a,e}, {a,d}, {a,h}, {b,d}, {b,f}.{b,e}, {c,g}, {c,e}, {e,g}.

Now from among these maximal independent set of vertices, we have to select the minimum number of such sets which can include all vertices of the graph and one such way is

{a,d} , {b,f} , {c,e}, {e,g}

This is the chromatic partitioning of this graph and since each one is an independent set we can assign a single color to color all the vertices within an independent set and we need 4 colors to color these 4 sets. So our graph only needs 4 colors for proper coloring

Hence, the chromatic number of this graph is 4.

You can refer to the below question and see the answer by Sukanya Das ma'am and Ayush Upadhyaya sir for more reference.

https://gateoverflow.in/204092/gate2018-18?show=204092#q204092

Ayush Upadhyaya Sir could you please check whether its the right approach or not?