# B theorem 24 page 34 proofs were also done in class

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(b): Theorem 2.4, page 34. Proofs were also done in class. These varied somewhat from those of the text. _________________________________________________________________ 9. (10 pts.) (a) Suppose G is a bipartite graph of order at least 5. Prove that the complement of G is not bipartite. [Hint: At least one partite set has three elements. Connect the dots?] Suppose that G is a bipartite graph of order at least 5 with partite sets U and W. At least one of U and W has at least 3 elements. Suppose without loss of generality, W 3. Label three of the members of W with u,v, and w. Since these vertices are in the same partite set of G, none of these three vertices is adjacent to any other of the three. Thus, uv , vw , uw E ( G ) G contains a 3 cycle . Thus, the complement of G is not bipartite. [Problem 1.25?] (b) Display a bipartite graph G of order 4 and its bipartite complement. Label each appropriately and give partite sets for each bipartite graph. //There are a multitude of examples. See me if you need help with this! This reveals why we want V(G) 5 in (a).
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