Pease prove the following e 2044Solutiona we are given that

Pease prove the following:

(e) 20->(4,4).

(a)

we are given that n,a,b >=2

and n is the number of vertice of the graph G

alpha(G)>=a and alpha(G) is the independence number

and omega(G)>=b

so if n>=2

and since n --> (a,b) is gien to us

=> n --> (2,2) would hold true

(b)

if n>=2

then a could be equal to n as we are given that n,a>=2

and since n -->(a,b) and n>=2

hence n --> (n , 2)

(c)

if n --> (a,b) and m >=n

since n , a, b >=2

so n=a=b =2

and m>=n

hence m could be = 2

and when that happens

m, a, b,>=2

and m --> (a,b)

hence proved

(d)

(m+n) --> ( 2a-1 , 2b-1 )

we know that a>=2 and b>= 2

=> (n+m) --> (4-1 , 4-1)

(n + m) --> (3,3)

we are given that a,b>=3

therefore (n+m) --> (a,b) holds true

(e)

since n , a , b >= 2

and since the graph G has n vrtices

so n coul be = 20

and when n=2

alpha (G) >=a

so a could be =4 and so could b = 4

therefore and we have already proved that

n --> (a,b)

therefore

20 --> (4,4) could be possible