Section 10.2 275 11. If i > j > 3, then VI V2Vj-l Vj is a 4-cycle. 111. If i = 4 and j = 3, then VI is adjacent to exactly half of V5, ..• ,Vn while V2 is adjacent to the other half. If V2 is adjacent to V5, choosing k > 5 minimal with V2 not adjacent to Vk, we have the 4-cycle VI V2Vk-l Vk. If VI is adjacent to V5, choose k > 5 minimal with VI and Vk not adjacent. Then VI V2VkVk-l is a 4-cycle. iv. If j = n and VI is adjacent to Vn-l, then VI V2Vn Vn-l is a 4-cycle. v. If j = n and V2 is adjacent to Vn-lo then VI V2Vn-l Vn is a 4-cycle. We note that the case j = n, i = n - 1 is not possible. VI. If n > j > i and VI is adjacent to Vj+l, then VI V2VjVj+! is a 4-cycle. vii. If n > j > i and V2 is adjacent to Vj+!, then VI V2 Vj+ 1 Vj is a 4-cycle. 21. (a) False. A Hamiltonian cycle in a graph contains no proper cycles. The graph itself, on the other hand, can be expected to contain many proper cycles, for example, any complete graph Kn (n > 3). (b) The complete graph
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