Question: (a) Let k Z+, k ¥ 3. If G = (V, E) is a connected planar graph with |V| = v, |E| = e, and

(a) Let k ˆˆ Z+, k ‰¥ 3. If G = (V, E) is a connected planar graph with |V| = v, |E| = e, and each cycle of length at least k, prove that

(b) What is the minimal cycle length in K3,3?
(c) Use parts (a) and (b) to conclude that K3, 3 is nonplanar.
(d) Use part (a) to prove that the Petersen graph is nonplanar.

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