# There are 3 proofs: 1) Let G be a simple graph with 2n (2 times n) vertices and

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There are 3 proofs:
1) Let G be a simple graph with 2n (2 times n) vertices and n^2 (n squared) edges. If G has no triangles, then G is the complete bipartite graph K_n,n (K sub n,n)
2) Prove that, if two distinct cycles of a graph G each contain an edge e, then G has a cycle that does not contain e.
3) Let G be a simple graph on 2k vertices containing no triangles. Prove, by induction on k, that G has at most k^2 (k squared) edges, and give an example of a graph for which this upper bound is achieved. (This result is often called Turán’s extremal theorem.) Use the proof from question 1 to help establish this problem.
For each proof complete the following:
– State the hypotheses
– State the conclusions
– Clearly and precisely prove the conclusions from the hypotheses
– Results presented earlier in the text may be used and must be clearly documented
– All equations must be done in the MS Word Equation editor