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MACM 101 — Discrete Mathematics I
Outline Solutions to Exercises on Graph Theory
1.
Use graph invariants to show that the following graphs are not isomorphic, or find a homomorphism
between them.
The graphs are isomorphic, and the following labeling of vertices gives an isomorphism.
a
b
c
d
e
f
g
h
a
b
c
d
e
f
g
h
2.
Show that every connected graph with
n
vertices has at least
n

1
edges.
By induction on the number of vertices. Let
P
(
n
)
denote the statement ‘Every connected graph with
n
vertices
has at least
n

1
edges’.
Basis step.
Verify
P
(2)
. Every connected graph with 2 vertices contains at least one edge, because otherwise it
is a pair of isolated vertices. Hence
P
(2)
is true.
Inductive step.
Suppose
P
(
m
)
is true for all
m
≤
k
; we prove
P
(
k
+ 1)
. Take any connected graph
G
with
k
+1
vertices and pick its vertex
v
. Let
G
′
be the graph obtained from
G
by removing
v
and all edges incident to
it.
G
′
can be disconnected, so let
G
1
, G
2
, . . . , G
s
be the connected components of
G
′
, and let
k
i
be the number
of vertices in
G
i
. By the inductive hypothesis, the number of edges in
G
i
is at least
k
i

1
, and the number of
edges in
G
′
is
(
k
1

1) + (
k
2

1) +
. . .
+ (
k
s

1) =
k

s.
Observe that there is an edge from
v
to some vertex in each component
G
i
, because otherwise
G
i
would be
disconnected from the rest of the graph
G
, as well, while
G
is connected. Therefore the number of edges in
G
is at least
(
k

s
) +
s
=
k
= (
k
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 Spring '09
 jcliu

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