answer already given in inmage

2 votes

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ANSWER are already given but still u want answer .

For 1 option we need 4 colors

For 2 option we need 3 colors

For 1 option we need 4 colors

For 2 option we need 3 colors

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Chromatic Number:- Minimum number of color needed to mark every vertices of a graph ,but no two adjacent vertices can have same color

QUESTION :1

Let B be RED

then W be Blue G can not be red or blue as then it will be two same color connecting let G YELLOW

R can be BLUE as it is connected with B and G which has no Blue Color(We could have marked with anyother color ,but we need minimum color number)

Right hand @W can again be YELLOW

Inthis way you can progress

QUESTION :1

Let B be RED

then W be Blue G can not be red or blue as then it will be two same color connecting let G YELLOW

R can be BLUE as it is connected with B and G which has no Blue Color(We could have marked with anyother color ,but we need minimum color number)

Right hand @W can again be YELLOW

Inthis way you can progress

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https://gateoverflow.in/204092/gate2018-18?show=206903#a206903

Read this answer and this comment:

https://gateoverflow.in/204092/gate2018-18?show=219861#c219861

Now for first image, minimum number of independent sets cover entire graph are 4. They are {a,d},{b,c},{g,f},{e,h}. So we need minimum of 4 colours to colour the graph.

For 2nd image, minimum number of independent sets that cover entire graph are 3. They are {a,d,f,h},{b,c,e,i},{g}. So we need minimum of 3 colours to colour the graph.

Read this answer and this comment:

https://gateoverflow.in/204092/gate2018-18?show=219861#c219861

Now for first image, minimum number of independent sets cover entire graph are 4. They are {a,d},{b,c},{g,f},{e,h}. So we need minimum of 4 colours to colour the graph.

For 2nd image, minimum number of independent sets that cover entire graph are 3. They are {a,d,f,h},{b,c,e,i},{g}. So we need minimum of 3 colours to colour the graph.