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after mid2 - 3 a For n E 2:1 a 2 show that the number of...

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Unformatted text preview: 3. a) For n E 2+. :1 a 2. show that the number of distinct Hamilton cycles in the graph KM is [1 HM}: — l}! a! b) How many different Hamilton paths are there fer K”, a I: l? 12. Prove that for n 3 2. the hypercube Q, has a Hamilton cycle. 13. Helen and Dominic invite It} friends to dinner. In this group of 12 people everyone knows at least 6 others. Prove that the 12 can be seated around a circular table in such a way that each person is acquainted with the persons sitting on either side. It]. Let G = (V, E} be a loop-free undirected n-regular graph With WI 2 2n + 2. PTove that? (the complement of G) has a Hamilton cycle. 3. If G is a loop-free undirected graph with at least one edge? prove that G is bipartite ifan onlv if xtG} = 2. 14. Let G be a loop-free undirected graph, where a. = maxugyidegtvfl. (a) PI‘Ove that KEG} 5 it + l. (b) Find two types of graphs Gt where 3.16) = a. +1. 15. For n E 3. let C... denote the cycle of length n. a) What is PHI}, It)? h) Ifrt E 4, show that 13(13le = PtPs—i, 1-) — P[Ce—la l], where PM. denotes the path of length n — l. c) Verify that P(P,,_], It] = MA. —1)"", for all n 3 2. 16. For n a 3, recall that the wheel graph, W". is obtained from a cycle of length n by placing a new vertex within the cycle and adding edges {spokes} from this new vertex to each vertex of the cycle. a) What relationship is there between 9: (CH) and 5:: (W5)? h} Use part [e] of Exercise IE to show that HIV”, 1.) = m — 2)“ + (—1)"i(i — 2}. ...
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