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
This is the end of the preview. Sign up
access the rest of the document.
Glossary of graph theory, hamiltonian cycle, Graph theory objects, Eulerian path, Hamiltonian path, Eulerian