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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 loopfree undirected nregular graph
With WI 2 2n + 2. PTove that? (the complement of G) has a
Hamilton cycle. 3. If G is a loopfree undirected graph with at least one edge?
prove that G is bipartite ifan onlv if xtG} = 2. 14. Let G be a loopfree 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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 Spring '08
 PEARCE

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