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
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 staff
 Math, pts, Natural number, Prime number, simple graph

Click to edit the document details