and append to it each of the closed paths to ensure that each
edge is used exactly once.
Exercise 1.5
Show that for every graph
G
, there exists a graph
H
so that
G
is a minor of
H
, and
for all
v
∈
V
(
H
), deg(
v
)
≤
3.
Proof.
Let
G
= (
V, E
) be a graph such that deg(
v
)
>
3 for every
v
∈
V
, and let
w
1
∈
V
be the vertex of
largest degree. Contract
G
by
w
1
to get
G
0
= (
V
0
, E
0
). If the degree of every vertex in
G
0
is still greater
than 3, choose the largest vertex
w
2
∈
V
0
and contract
G
0
by
w
2
to get
G
00
= (
V
00
, E
00
). Proceed in this
manner until the degree of every vertex is at most 3. This is clearly possible, because one can simply go
contracting off vertices until there insufficiently many of them for any to have degree greater than 3.
Exercise 1.11
Recall that
b
0
(
G
) is the number of connected components of a graph
G
. Show that
b
0
(
G
) = dim
V 
dim Im(
d
)
Lemma
Let
G
= (
V, E
) be a graph. If
G
has no cycles, then
G
has

V
  
E

connected components.
Proof.
Since
G
has no cycles,
G
consists of (possibly disconnected) trees, meaning that each connected
component
G
i
= (
V
i
, E
i
) of
G
satisfies

E
i

=

V
i
 
1, or 1 =

V
i
  
E
i

.
Summing over all of the
connected components gives
k
=

V
  
E

, so
G
has
k
connected components.
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 Spring '08
 Staff
 Math