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184a_98w_1e

# 184a_98w_1e - Math 184A First Hour Exam 30 January 1998 1(6...

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Math 184A First Hour Exam 30 January 1998 1. (6 pts.) Recall that a cycle of a simple graph is a subgraph consisting of some set { v 1 , . . . , v k } of vertices together with the k edges { v 1 , v 2 } , { v 2 , v 3 } , . . . , { v k , v 1 } . It follows from Exercise 5.5.3 that a connected v -vertex graph that has at least one cycle has at least v edges. (You are not asked to do that.) It can be shown that a connected v -vertex graph that has at least two cycles has at least v + 1 edges. (You are not asked to do that.) One might expect the pattern to continue: at least k cycles implies at least v + k - 1 edges. Show that this doesn’t happen by exhibiting for some v a connected v -vertex simple graph with more than two cycles and only v + 1 edges . Be sure to describe the cycles! 2. (a) (8 pts.) How many ways are there to form an ordered/ list of 3 (three) letters from the letters in LAJOLLA (3 L’s, 2 A’s, 1 J, and 1 O), provided no letter can be used
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