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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
3
}
,
...
,
{
v
k
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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This note was uploaded on 06/18/2009 for the course MATH 184a taught by Professor Staff during the Winter '08 term at UCSD.
 Winter '08
 staff
 Math

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