Question

Problem 1. Consider a graph G with n vertices: • Ifq is the size of the largest independent set in graph G, show that ax(G) n. • Use the previous result to prove that x(G)(n - d) > n and thuse x(G) > n/(n - d), where d is the minimal degree of the vertex in G. • Prove that x(G) + X(G) Sn +1 (Hint: Induction). • x(G)x(G) 2 n. • Use previous inequality to show that x(G) + x(G) > 2/n.

Answer 1

Hello. I understand that you wanted all your questions to have solved. But Chegg Expert Policy doesn't allow us to solve more than one question. So please understand that even as Chegg expert we have limitations too.
I have provided detailed answer but if you have doubts please ask. I'd love to help.
It's a humble request to please upvote the answer. I really need this help now. Please.
Have a nice day.
Ginen. a) According to question q is the size of the largest independent set in a graph G - To n., where n is number in of y (4) ► matices in G. This can be pro ned by Contradiction, which me assume xchlq cn. what this equation means Color (any one) , no of vertices with that colour will be mime than the value of a because solne x(algan : x (ok n Now we know about or is the largest independent set a so there is a conmadichono tenu our assump- ton is wrong so we Txalq n it we conclude (6) Ginen - ni numrin degree = d. Moiry dtq in size of largest independent Set. here q is d+q>n d+q-dan-d. using now S solning Scanned with Cay(G) a 3 n& putting q = nod. we get 1X (G) ha)

with investices. and let G be a graph of a is complete then x(u) an x (4)=1 of G is not complete then, x(h) cm xC6) al G with graph in nestices. Therefore, for any a (4) + x(u) snu CS Scanned with CamScanner

Let Ghia 24ans aguaph. Lut J1G)=k X(G)=d. We start with kn compute quaph on Orecties Noue to each e veuten gG (or Go one anoeiate the. oudred pain ((0) (0)}. Giren reutius (dishnet) e and win an uis adjacent to us in theu Gorg almon-ke woning of colis > N(44) Skl EXIG).XIGC) DE X(G) XGC). Es bing proce) en: so this s gives aide KA using
according to HomeworkLib policy i answered first four question.sry i can't answer more.