Question

Graphic Theory Question: Will upvote all answers.

Please read carefully and answer clearly (easy to read).

Question 5. Consider the statement, "If G is a graph of order at least 5, then at most one of G and G is bipartite" Here is a picture of your book's proof: 1.25 Proof. If G is not bipartite, then we have the desired result. Thus, we may assume that G is bipartite. Let i and V2 be two bipartite sets of G. Since the order of G is at least 5, at least one of Vi and V2 contains 3 or more vertices say 3. Since the subgraph of G induced by V is the complete graph K and k 2 3, it follows that G contains a triangle. By Theorem 1.12, G is not bipartite. Answer the following questions about the proof: (a) What kind of statement is this? (Universal or existential?) b) What kind of proof is this? (Induction? Direct? Contradiction?) (c) The proof does not explicitly state what it is assuming. Write a better first sentence of the proof that DOES state the assumptions. (d) Explain the logic behind the first sentence written in this proof. (e) Explain why the statement labeled with a red " is true (f) Explain why the statement labeled with a red "2" is true (g) What is Theorem 1.12, and how does it help? (h) Where does the proof use the fact that the order is at least 5? Is the statement true when the order is 47

Theorem 1.12) A nontrivial graph G is a bipartite graph if and only if G contains no odd cycles.

Answer 1

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