Question: 1.Prove that if G = (V, E) is k-colourable then G has a vertex cover of size n dn k e, where n = |V

1.Prove that if G = (V, E) is k-colourable then G has a vertex cover of size n dn k e, where n = |V |. Use a direct proof where each step follows by implication from definitions, known facts, and the previous lines.

2.Clearly state the converse of the claim in a. Prove it is not true by giving a disproof by counter-example of the converse. Please draw your graphs so that the markers can understand them (photos as pdfs are okay).

3."Reverse" your proof from part a. (Since it was a series of implications, this should be easy to dojust start at the end and imply backwards.) We know from part b. that your reversed proof cannot possibly hold. Which step(s) of your reversed proof fails and why?

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