6.042/18.062J
Mathematics
for
Computer
Science
September
28,
2010
Tom
Leighton
and
Marten
van
Dijk
Problem
Set
4
Problem
1.
[15
points]
Let
G
= (
V, E
)
be
a
graph.
A
matching
in
G
is
a
set
M
⊂
E
such
that
no
two
edges
in
M
are
incident
on
a
common
vertex.
Let
M
1
,
M
2
be
two
matchings
of
G
.
Consider
the
new
graph
G
�
= (
V, M
1
∪
M
2
)
(i.e.
on
the
same
vertex
set,
whose
edges
consist
of
all
the
edges
that
appear
in
either
M
1
or
M
2
).
Show
that
G
�
is
bipartite.
Helpful
definition
: A
connected
component
is
a
subgraph
of
a
graph
consisting
of
some
vertex
and
every
node
and
edge
that
is
connected
to
that
vertex.
Problem
2.
[20
points]
Let
G
= (
V, E
)
be
a
graph.
Recall
that
the
degree
of
a
vertex
v
∈
V
,
denoted
d
v
,
is
the
number
of
vertices
w
such
that
there
is
an
edge
between
v
and
w
.
(a)
[10
pts]
Prove
that
�
2

E

=
d
v
.
v
∈
V
(b)
[5
pts]
At
a
6.042
ice
cream
study
session
(where
the
ice
cream
is
plentiful
and
it
helps
you
study
too)
111
students
showed
up.
During
the
session,
some
students
shook
hands
with
each
other
(everybody
being
happy
and
content
with
the
icecream
and
all).
Turns
out
that
the
University
of
Chicago
did
another
spectacular
study
here,
and
counted
that
each
student
shook
hands
with
exactly
17
other
students.
Can
you
debunk
this
too?
(c)
[5
pts]
And
on
a
more
dull
note,
how
many
edges
does
K
n
,
the
complete
graph
on
n
vertices,
have?
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 Fall '10
 TomLeighton,Dr.MartenvanDijk
 Computer Science, Graph Theory, Vertex, line graph, vertices

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