Question

Let G and H be the graphs find a)\({x_2}\left( G \right)\) b) \({x_2}\left( H \right)\) c)\({x_3}\left( H \right)\) d)\({x_2}\left( H \right)\)

Short answer

1. The value of\({x_2}\left( G \right)\)is 6
2. The value of\({x_2}\left( H \right)\)is 7
3. The value of \({x_3}\left( G \right)\)is
4. The value of \({x_3}\left( H \right)\)is 11

Step-by-step solution

Given information

Given are the\({x_2}\left( G \right)\), \({x_2}\left( H \right)\),\({x_3}\left( G \right)\) ,\({x_3}\left( H \right)\)

Concept

\({x_m}\left( G \right)\)Represents\(m\)

Solution part a

Here, we can assign color to vertices as

\(\begin{array}{l}Col\left( a \right) = \left( {1,2} \right)\\Col\left( b \right) = \left( {3,4} \right)\\Col\left( c \right) = \left( {5,6} \right)\\Col\left( d \right) = \left( {1,2} \right)\\Col\left( e \right) = \left( {5,6} \right)\\Col\left( f \right) = \left( {3,4} \right)\\Col\left( g \right) = \left( {1,2} \right)\end{array}\)

Therefore, the value of \({x_2}\left( G \right)\)=\(6\)

Solution part b

Here, we can assign color to vertices as

\(\begin{array}{l}Col\left( a \right) = \left( {1,2} \right)\\Col\left( b \right) = \left( {3,4} \right)\\Col\left( c \right) = \left( {5,6} \right)\\Col\left( d \right) = \left( {1,2} \right)\\Col\left( e \right) = \left( {5,6} \right)\\Col\left( f \right) = \left( {3,4} \right)\\Col\left( g \right) = \left( {1,7} \right)\end{array}\)

Therefore, the value of \({x_3}\left( G \right)\)=\(7\)

Solution part c

Here, we can assign color to vertices as

\(\begin{array}{l}Col\left( a \right) = \left( {1,2,7} \right)\\Col\left( b \right) = \left( {3,4,8} \right)\\Col\left( c \right) = \left( {5,6,9} \right)\\Col\left( d \right) = \left( {1,2,7} \right)\\Col\left( e \right) = \left( {5,6,9} \right)\\Col\left( f \right) = \left( {3,4,8} \right)\\Col\left( g \right) = \left( {1,2,7} \right)\end{array}\)

Therefore, the value of \({x_2}\left( H \right)\)=9

Solution part d

\(\begin{array}{l}Col\left( a \right) = \left( {1,2,7} \right)\\Col\left( b \right) = \left( {3,4,8} \right)\\Col\left( c \right) = \left( {5,6,9} \right)\\Col\left( d \right) = \left( {1,2,7} \right)\\Col\left( e \right) = \left( {5,6,9} \right)\\Col\left( f \right) = \left( {3,4,8} \right)\\Col\left( g \right) = \left( {1,10,11} \right)\end{array}\)

The final values of the following are:

1. The value of\({x_2}\left( G \right)\)is 6
2. The value of\({x_2}\left( H \right)\)is 7
3. The value of\({x_3}\left( G \right)\)is 9
4. The value of \({x_3}\left( H \right)\)is 11