1.
Write the graphs whose pictures appear in the text as pairs of sets
(
V; E
)
.
(a)
(
f
1
;
2
;
3
;
4
;
5
;
6
g
;
ff
1
;
2
g
;
f
1
;
4
g
;
f
2
;
3
g
;
f
2
;
5
g
;
f
3
;
6
g
;
f
4
;
5
g
;
f
5
;
6
gg
)
(b)
(
f
1
;
2
;
3
;
4
;
5
;
6
g
;
ff
1
;
2
g
;
f
1
;
4
g
;
f
1
;
6
g
;
f
2
;
3
g
;
f
2
;
5
g
;
f
3
;
4
g
;
f
4
;
5
g
;
f
4
;
6
g
;
f
5
;
6
gg
)
(c)
(
f
1
;
2
;
3
;
4
;
5
;
6
g
;
ff
1
;
2
g
;
f
1
;
4
g
;
f
2
;
5
g
;
f
4
;
5
gg
)
2.
Draw pictures of
(a)
(
f
a; b; c; d; e
g
;
ff
a; b
g
;
f
a; c
g
;
f
a; d
g
;
f
b; e
g
;
f
c; d
gg
)
e
e
e
e
e
c
a
b
e
d
(b)
(
f
a; b; c; d; e
g
;
ff
a; b
g
;
f
a; c
g
;
f
b; c
g
;
f
b; d
g
;
f
c; d
gg
)
e
e
e
e
e
±
±
±
±
±
±
a
b
d
e
c
(c)
(
f
a; b; c; d; e
g
;
ff
a; c
g
;
f
b; d
g
;
f
b; e
gg
)
e
e
e
e
e
a
c
d
b
e
12.
Let
G
be a graph. Prove that there must be an even number of vertices
of odd degree
.
If there were an odd number of vertices with odd degree, then the sum
of the degrees of all the vertices would be odd. But Theorem 46.5 says
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 Spring '08
 STAFF
 Graph Theory, 1g, vertices

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